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Vincent Selhorst-Jones

Vincent Selhorst-Jones

Inverse Functions

Slide Duration:

Table of Contents

I. Introduction
Introduction to Math Analysis

10m 3s

Intro
0:00
Title of the Course
0:06
Different Names for the Course
0:07
Precalculus
0:12
Math Analysis
0:14
Trigonometry
0:16
Algebra III
0:20
Geometry II
0:24
College Algebra
0:30
Same Concepts
0:36
How do the Lessons Work?
0:54
Introducing Concepts
0:56
Apply Concepts
1:04
Go through Examples
1:25
Who is this Course For?
1:38
Those Who Need eExtra Help with Class Work
1:52
Those Working on Material but not in Formal Class at School
1:54
Those Who Want a Refresher
2:00
Try to Watch the Whole Lesson
2:20
Understanding is So Important
3:56
What to Watch First
5:26
Lesson #2: Sets, Elements, and Numbers
5:30
Lesson #7: Idea of a Function
5:33
Lesson #6: Word Problems
6:04
What to Watch First, cont.
6:46
Lesson #2: Sets, Elements and Numbers
6:56
Lesson #3: Variables, Equations, and Algebra
6:58
Lesson #4: Coordinate Systems
7:00
Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope
7:02
Lesson #6: Word Problems
7:10
Lesson #7: Idea of a Function
7:12
Lesson #8: Graphs
7:14
Graphing Calculator Appendix
7:40
What to Watch Last
8:46
Let's get Started!
9:48
Sets, Elements, & Numbers

45m 11s

Intro
0:00
Introduction
0:05
Sets and Elements
1:19
Set
1:20
Element
1:23
Name a Set
2:20
Order The Elements Appear In Has No Effect on the Set
2:55
Describing/ Defining Sets
3:28
Directly Say All the Elements
3:36
Clearly Describing All the Members of the Set
3:55
Describing the Quality (or Qualities) Each member Of the Set Has In Common
4:32
Symbols: 'Element of' and 'Subset of'
6:01
Symbol is ∈
6:03
Subset Symbol is ⊂
6:35
Empty Set
8:07
Symbol is ∅
8:20
Since It's Empty, It is a Subset of All Sets
8:44
Union and Intersection
9:54
Union Symbol is ∪
10:08
Intersection Symbol is ∩
10:18
Sets Can Be Weird Stuff
12:26
Can Have Elements in a Set
12:50
We Can Have Infinite Sets
13:09
Example
13:22
Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
14:08
This Set Has Infinitely Many Distinct Elements
14:40
Numbers as Sets
16:03
Natural Numbers ℕ
16:16
Including 0 and the Negatives ℤ
18:13
Rational Numbers ℚ
19:27
Can Express Rational Numbers with Decimal Expansions
22:05
Irrational Numbers
23:37
Real Numbers ℝ: Put the Rational and Irrational Numbers Together
25:15
Interval Notation and the Real Numbers
26:45
Include the End Numbers
27:06
Exclude the End Numbers
27:33
Example
28:28
Interval Notation: Infinity
29:09
Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
29:14
Always Use Parentheses
29:50
Examples
30:27
Example 1
31:23
Example 2
35:26
Example 3
38:02
Example 4
42:21
Variables, Equations, & Algebra

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
Put Your Substitution in Parentheses
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

Intro
0:00
Inherent Order in ℝ
0:05
Real Numbers Come with an Inherent Order
0:11
Positive Numbers
0:21
Negative Numbers
0:58
'Less Than' and 'Greater Than'
2:04
Tip To Help You Remember the Signs
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
The Plane and Quadrants
18:50
Quadrant I
19:04
Quadrant II
19:21
Quadrant III
20:04
Quadrant IV
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

Intro
0:00
Introduction
0:07
Midpoint: One Dimension
2:09
Example of Something More Complex
2:31
Use the Idea of a Middle
3:28
Find the Midpoint of Arbitrary Values a and b
4:17
How They're Equivalent
5:05
Official Midpoint Formula
5:46
Midpoint: Two Dimensions
6:19
The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
6:38
Arbitrary Pair of Points Example
7:25
Distance: One Dimension
9:26
Absolute Value
10:54
Idea of Forcing Positive
11:06
Distance: One Dimension, Formula
11:47
Distance Between Arbitrary a and b
11:48
Absolute Value Helps When the Distance is Negative
12:41
Distance Formula
12:58
The Pythagorean Theorem
13:24
a²+b²=c²
13:50
Distance: Two Dimensions
14:59
Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
15:16
Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
16:21
Slope
19:30
Slope is the Rate of Change
19:41
m = rise over run
21:27
Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
22:31
Interpreting Slope
24:12
Positive Slope and Negative Slope
25:40
m=1, m=0, m=-1
26:48
Example 1
28:25
Example 2
31:42
Example 3
36:40
Example 4
42:48
Word Problems

56m 31s

Intro
0:00
Introduction
0:05
What is a Word Problem?
0:45
Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
0:48
Requires Us to Think
1:32
Why Are They So Hard?
2:11
Reason 1: No Simple Formula to Solve Them
2:16
Reason 2: Harder to Teach Word Problems
2:47
You Can Learn How to Do Them!
3:51
Grades
7:57
'But I'm Never Going to Use This In Real Life'
9:46
Solving Word Problems
12:58
First: Understand the Problem
13:37
Second: What Are You Looking For?
14:33
Third: Set Up Relationships
16:21
Fourth: Solve It!
17:48
Summary of Method
19:04
Examples on Things Other Than Math
20:21
Math-Specific Method: What You Need Now
25:30
Understand What the Problem is Talking About
25:37
Set Up and Name Any Variables You Need to Know
25:56
Set Up Equations Connecting Those Variables to the Information in the Problem Statement
26:02
Use the Equations to Solve for an Answer
26:14
Tip
26:58
Draw Pictures
27:22
Breaking Into Pieces
28:28
Try Out Hypothetical Numbers
29:52
Student Logic
31:27
Jump In!
32:40
Example 1
34:03
Example 2
39:15
Example 3
44:22
Example 4
50:24
II. Functions
Idea of a Function

39m 54s

Intro
0:00
Introduction
0:04
What is a Function?
1:06
A Visual Example and Non-Example
1:30
Function Notation
3:47
f(x)
4:05
Express What Sets the Function Acts On
5:45
Metaphors for a Function
6:17
Transformation
6:28
Map
7:17
Machine
8:56
Same Input Always Gives Same Output
10:01
If We Put the Same Input Into a Function, It Will Always Produce the Same Output
10:11
Example of Something That is Not a Function
11:10
A Non-Numerical Example
12:10
The Functions We Will Use
15:05
Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers
15:11
Usually Told the Rule of a Given Function
15:27
How To Use a Function
16:18
Apply the Rule to Whatever Our Input Value Is
16:28
Make Sure to Wrap Your Substitutions in Parentheses
17:09
Functions and Tables
17:36
Table of Values, Sometimes Called a T-Table
17:46
Example
17:56
Domain: What Goes In
18:55
The Domain is the Set of all Inputs That the Function Can Accept
18:56
Example
19:40
Range: What Comes Out
21:27
The Range is the Set of All Possible Outputs a Function Can Assign
21:34
Example
21:49
Another Example Would Be Our Initial Function From Earlier in This Lesson
22:29
Example 1
23:45
Example 2
25:22
Example 3
27:27
Example 4
29:23
Example 5
33:33
Graphs

58m 26s

Intro
0:00
Introduction
0:04
How to Interpret Graphs
1:17
Input / Independent Variable
1:47
Output / Dependent Variable
2:00
Graph as Input ⇒ Output
2:23
One Way to Think of a Graph: See What Happened to Various Inputs
2:25
Example
2:47
Graph as Location of Solution
4:20
A Way to See Solutions
4:36
Example
5:20
Which Way Should We Interpret?
7:13
Easiest to Think In Terms of How Inputs Are Mapped to Outputs
7:20
Sometimes It's Easier to Think In Terms of Solutions
8:39
Pay Attention to Axes
9:50
Axes Tell Where the Graph Is and What Scale It Has
10:09
Often, The Axes Will Be Square
10:14
Example
12:06
Arrows or No Arrows?
16:07
Will Not Use Arrows at the End of Our Graphs
17:13
Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
17:18
How to Graph
19:47
Plot Points
20:07
Connect with Curves
21:09
If You Connect with Straight Lines
21:44
Graphs of Functions are Smooth
22:21
More Points ⇒ More Accurate
23:38
Vertical Line Test
27:44
If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function
28:41
Every Point on a Graph Tells Us Where the x-Value Below is Mapped
30:07
Domain in Graphs
31:37
The Domain is the Set of All Inputs That a Function Can Accept
31:44
Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
33:19
Range in Graphs
33:53
Graphing Calculators: Check the Appendix!
36:55
Example 1
38:37
Example 2
45:19
Example 3
50:41
Example 4
53:28
Example 5
55:50
Properties of Functions

48m 49s

Intro
0:00
Introduction
0:05
Increasing Decreasing Constant
0:43
Looking at a Specific Graph
1:15
Increasing Interval
2:39
Constant Function
4:15
Decreasing Interval
5:10
Find Intervals by Looking at the Graph
5:32
Intervals Show x-values; Write in Parentheses
6:39
Maximum and Minimums
8:48
Relative (Local) Max/Min
10:20
Formal Definition of Relative Maximum
12:44
Formal Definition of Relative Minimum
13:05
Max/Min, More Terms
14:18
Definition of Extrema
15:01
Average Rate of Change
16:11
Drawing a Line for the Average Rate
16:48
Using the Slope of the Secant Line
17:36
Slope in Function Notation
18:45
Zeros/Roots/x-intercepts
19:45
What Zeros in a Function Mean
20:25
Even Functions
22:30
Odd Functions
24:36
Even/Odd Functions and Graphs
26:28
Example of an Even Function
27:12
Example of an Odd Function
28:03
Example 1
29:35
Example 2
33:07
Example 3
40:32
Example 4
42:34
Function Petting Zoo

29m 20s

Intro
0:00
Introduction
0:04
Don't Forget that Axes Matter!
1:44
The Constant Function
2:40
The Identity Function
3:44
The Square Function
4:40
The Cube Function
5:44
The Square Root Function
6:51
The Reciprocal Function
8:11
The Absolute Value Function
10:19
The Trigonometric Functions
11:56
f(x)=sin(x)
12:12
f(x)=cos(x)
12:24
Alternate Axes
12:40
The Exponential and Logarithmic Functions
13:35
Exponential Functions
13:44
Logarithmic Functions
14:24
Alternating Axes
15:17
Transformations and Compositions
16:08
Example 1
17:52
Example 2
18:33
Example 3
20:24
Example 4
26:07
Transformation of Functions

48m 35s

Intro
0:00
Introduction
0:04
Vertical Shift
1:12
Graphical Example
1:21
A Further Explanation
2:16
Vertical Stretch/Shrink
3:34
Graph Shrinks
3:46
Graph Stretches
3:51
A Further Explanation
5:07
Horizontal Shift
6:49
Moving the Graph to the Right
7:28
Moving the Graph to the Left
8:12
A Further Explanation
8:19
Understanding Movement on the x-axis
8:38
Horizontal Stretch/Shrink
12:59
Shrinking the Graph
13:40
Stretching the Graph
13:48
A Further Explanation
13:55
Understanding Stretches from the x-axis
14:12
Vertical Flip (aka Mirror)
16:55
Example Graph
17:07
Multiplying the Vertical Component by -1
17:18
Horizontal Flip (aka Mirror)
18:43
Example Graph
19:01
Multiplying the Horizontal Component by -1
19:54
Summary of Transformations
22:11
Stacking Transformations
24:46
Order Matters
25:20
Transformation Example
25:52
Example 1
29:21
Example 2
34:44
Example 3
38:10
Example 4
43:46
Composite Functions

33m 24s

Intro
0:00
Introduction
0:04
Arithmetic Combinations
0:40
Basic Operations
1:20
Definition of the Four Arithmetic Combinations
1:40
Composite Functions
2:53
The Function as a Machine
3:32
Function Compositions as Multiple Machines
3:59
Notation for Composite Functions
4:46
Two Formats
6:02
Another Visual Interpretation
7:17
How to Use Composite Functions
8:21
Example of on Function acting on Another
9:17
Example 1
11:03
Example 2
15:27
Example 3
21:11
Example 4
27:06
Piecewise Functions

51m 42s

Intro
0:00
Introduction
0:04
Analogies to a Piecewise Function
1:16
Different Potatoes
1:41
Factory Production
2:27
Notations for Piecewise Functions
3:39
Notation Examples from Analogies
6:11
Example of a Piecewise (with Table)
7:24
Example of a Non-Numerical Piecewise
11:35
Graphing Piecewise Functions
14:15
Graphing Piecewise Functions, Example
16:26
Continuous Functions
16:57
Statements of Continuity
19:30
Example of Continuous and Non-Continuous Graphs
20:05
Interesting Functions: the Step Function
22:00
Notation for the Step Function
22:40
How the Step Function Works
22:56
Graph of the Step Function
25:30
Example 1
26:22
Example 2
28:49
Example 3
36:50
Example 4
46:11
Inverse Functions

49m 37s

Intro
0:00
Introduction
0:04
Analogy by picture
1:10
How to Denote the inverse
1:40
What Comes out of the Inverse
1:52
Requirement for Reversing
2:02
The Basketball Factory
2:12
The Importance of Information
2:45
One-to-One
4:04
Requirement for Reversibility
4:21
When a Function has an Inverse
4:43
One-to-One
5:13
Not One-to-One
5:50
Not a Function
6:19
Horizontal Line Test
7:01
How to the test Works
7:12
One-to-One
8:12
Not One-to-One
8:45
Definition: Inverse Function
9:12
Formal Definition
9:21
Caution to Students
10:02
Domain and Range
11:12
Finding the Range of the Function Inverse
11:56
Finding the Domain of the Function Inverse
12:11
Inverse of an Inverse
13:09
Its just x!
13:26
Proof
14:03
Graphical Interpretation
17:07
Horizontal Line Test
17:20
Graph of the Inverse
18:04
Swapping Inputs and Outputs to Draw Inverses
19:02
How to Find the Inverse
21:03
What We Are Looking For
21:21
Reversing the Function
21:38
A Method to Find Inverses
22:33
Check Function is One-to-One
23:04
Swap f(x) for y
23:25
Interchange x and y
23:41
Solve for y
24:12
Replace y with the inverse
24:40
Some Comments
25:01
Keeping Step 2 and 3 Straight
25:44
Switching to Inverse
26:12
Checking Inverses
28:52
How to Check an Inverse
29:06
Quick Example of How to Check
29:56
Example 1
31:48
Example 2
34:56
Example 3
39:29
Example 4
46:19
Variation Direct and Inverse

28m 49s

Intro
0:00
Introduction
0:06
Direct Variation
1:14
Same Direction
1:21
Common Example: Groceries
1:56
Different Ways to Say that Two Things Vary Directly
2:28
Basic Equation for Direct Variation
2:55
Inverse Variation
3:40
Opposite Direction
3:50
Common Example: Gravity
4:53
Different Ways to Say that Two Things Vary Indirectly
5:48
Basic Equation for Indirect Variation
6:33
Joint Variation
7:27
Equation for Joint Variation
7:53
Explanation of the Constant
8:48
Combined Variation
9:35
Gas Law as a Combination
9:44
Single Constant
10:33
Example 1
10:49
Example 2
13:34
Example 3
15:39
Example 4
19:48
III. Polynomials
Intro to Polynomials

38m 41s

Intro
0:00
Introduction
0:04
Definition of a Polynomial
1:04
Starting Integer
2:06
Structure of a Polynomial
2:49
The a Constants
3:34
Polynomial Function
5:13
Polynomial Equation
5:23
Polynomials with Different Variables
5:36
Degree
6:23
Informal Definition
6:31
Find the Largest Exponent Variable
6:44
Quick Examples
7:36
Special Names for Polynomials
8:59
Based on the Degree
9:23
Based on the Number of Terms
10:12
Distributive Property (aka 'FOIL')
11:37
Basic Distributive Property
12:21
Distributing Two Binomials
12:55
Longer Parentheses
15:12
Reverse: Factoring
17:26
Long-Term Behavior of Polynomials
17:48
Examples
18:13
Controlling Term--Term with the Largest Exponent
19:33
Positive and Negative Coefficients on the Controlling Term
20:21
Leading Coefficient Test
22:07
Even Degree, Positive Coefficient
22:13
Even Degree, Negative Coefficient
22:39
Odd Degree, Positive Coefficient
23:09
Odd Degree, Negative Coefficient
23:27
Example 1
25:11
Example 2
27:16
Example 3
31:16
Example 4
34:41
Roots (Zeros) of Polynomials

41m 7s

Intro
0:00
Introduction
0:05
Roots in Graphs
1:17
The x-intercepts
1:33
How to Remember What 'Roots' Are
1:50
Naïve Attempts
2:31
Isolating Variables
2:45
Failures of Isolating Variables
3:30
Missing Solutions
4:59
Factoring: How to Find Roots
6:28
How Factoring Works
6:36
Why Factoring Works
7:20
Steps to Finding Polynomial Roots
9:21
Factoring: How to Find Roots CAUTION
10:08
Factoring is Not Easy
11:32
Factoring Quadratics
13:08
Quadratic Trinomials
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
Factoring Quadratics, Check Your Work
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
Factoring a Quadratic
19:04
Factoring: Roots Imply Factors
19:54
Where a Root is, A Factor Is
20:01
How to Use Known Roots to Make Factoring Easier
20:35
Not all Polynomials Can be Factored
22:30
Irreducible Polynomials
23:27
Complex Numbers Help
23:55
Max Number of Roots/Factors
24:57
Limit to Number of Roots Equal to the Degree
25:18
Why there is a Limit
25:25
Max Number of Peaks/Valleys
26:39
Shape Information from Degree
26:46
Example Graph
26:54
Max, But Not Required
28:00
Example 1
28:37
Example 2
31:21
Example 3
36:12
Example 4
38:40
Completing the Square and the Quadratic Formula

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
Solving Quadratics with Ease
9:56
The Quadratic Formula
11:38
Derivation
11:43
Final Form
12:23
Follow Format to Use Formula
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
Example 2: Solve the Quadratic
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52
Properties of Quadratic Functions

45m 34s

Intro
0:00
Introduction
0:05
Parabolas
0:35
Examples of Different Parabolas
1:06
Axis of Symmetry and Vertex
1:28
Drawing an Axis of Symmetry
1:51
Placing the Vertex
2:28
Looking at the Axis of Symmetry and Vertex for other Parabolas
3:09
Transformations
4:18
Reviewing Transformation Rules
6:28
Note the Different Horizontal Shift Form
7:45
An Alternate Form to Quadratics
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

46m 8s

Intro
0:00
Introduction
0:05
Reminder: Roots Imply Factors
1:32
The Intermediate Value Theorem
3:41
The Basis: U between a and b
4:11
U is on the Function
4:52
Intermediate Value Theorem, Proof Sketch
5:51
If Not True, the Graph Would Have to Jump
5:58
But Graph is Defined as Continuous
6:43
Finding Roots with the Intermediate Value Theorem
7:01
Picking a and b to be of Different Signs
7:10
Must Be at Least One Root
7:46
Dividing a Polynomial
8:16
Using Roots and Division to Factor
8:38
Long Division Refresher
9:08
The Division Algorithm
12:18
How It Works to Divide Polynomials
12:37
The Parts of the Equation
13:24
Rewriting the Equation
14:47
Polynomial Long Division
16:20
Polynomial Long Division In Action
16:29
One Step at a Time
20:51
Synthetic Division
22:46
Setup
23:11
Synthetic Division, Example
24:44
Which Method Should We Use
26:39
Advantages of Synthetic Method
26:49
Advantages of Long Division
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

Intro
0:00
Introduction
0:04
A Wacky Idea
1:02
The Definition of the Imaginary Number
1:22
How it Helps Solve Equations
2:20
Square Roots and Imaginary Numbers
3:15
Complex Numbers
5:00
Real Part and Imaginary Part
5:20
When Two Complex Numbers are Equal
6:10
Addition and Subtraction
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
Factoring So-Called 'Irreducible' Quadratics
19:24
Revisiting the Quadratic Formula
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

19m 9s

Intro
0:00
Introduction
0:05
Idea: Hidden Roots
1:16
Roots in Complex Form
1:42
All Polynomials Have Roots
2:08
Fundamental Theorem of Algebra
2:21
Where Are All the Imaginary Roots, Then?
3:17
All Roots are Complex
3:45
Real Numbers are a Subset of Complex Numbers
3:59
The n Roots Theorem
5:01
For Any Polynomial, Its Degree is Equal to the Number of Roots
5:11
Equivalent Statement
5:24
Comments: Multiplicity
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
Comments: Complex Numbers Necessary
7:41
Comments: Complex Coefficients Allowed
8:55
Comments: Existence Theorem
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
IV. Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

Intro
0:00
Introduction
0:05
Definition of a Rational Function
1:20
Examples of Rational Functions
2:30
Why They are Called 'Rational'
2:47
Domain of a Rational Function
3:15
Undefined at Denominator Zeros
3:25
Otherwise all Reals
4:16
Investigating a Fundamental Function
4:50
The Domain of the Function
5:04
What Occurs at the Zeroes of the Denominator
5:20
Idea of a Vertical Asymptote
6:23
What's Going On?
6:58
Approaching x=0 from the left
7:32
Approaching x=0 from the right
8:34
Dividing by Very Small Numbers Results in Very Large Numbers
9:31
Definition of a Vertical Asymptote
10:05
Vertical Asymptotes and Graphs
11:15
Drawing Asymptotes by Using a Dashed Line
11:27
The Graph Can Never Touch Its Undefined Point
12:00
Not All Zeros Give Asymptotes
13:02
Special Cases: When Numerator and Denominator Go to Zero at the Same Time
14:58
Cancel out Common Factors
15:49
How to Find Vertical Asymptotes
16:10
Figure out What Values Are Not in the Domain of x
16:24
Determine if the Numerator and Denominator Share Common Factors and Cancel
16:45
Find Denominator Roots
17:33
Note if Asymptote Approaches Negative or Positive Infinity
18:06
Example 1
18:57
Example 2
21:26
Example 3
23:04
Example 4
30:01
Horizontal Asymptotes

34m 16s

Intro
0:00
Introduction
0:05
Investigating a Fundamental Function
0:53
What Happens as x Grows Large
1:00
Different View
1:12
Idea of a Horizontal Asymptote
1:36
What's Going On?
2:24
What Happens as x Grows to a Large Negative Number
2:49
What Happens as x Grows to a Large Number
3:30
Dividing by Very Large Numbers Results in Very Small Numbers
3:52
Example Function
4:41
Definition of a Vertical Asymptote
8:09
Expanding the Idea
9:03
What's Going On?
9:48
What Happens to the Function in the Long Run?
9:51
Rewriting the Function
10:13
Definition of a Slant Asymptote
12:09
Symbolical Definition
12:30
Informal Definition
12:45
Beyond Slant Asymptotes
13:03
Not Going Beyond Slant Asymptotes
14:39
Horizontal/Slant Asymptotes and Graphs
15:43
How to Find Horizontal and Slant Asymptotes
16:52
How to Find Horizontal Asymptotes
17:12
Expand the Given Polynomials
17:18
Compare the Degrees of the Numerator and Denominator
17:40
How to Find Slant Asymptotes
20:05
Slant Asymptotes Exist When n+m=1
20:08
Use Polynomial Division
20:24
Example 1
24:32
Example 2
25:53
Example 3
26:55
Example 4
29:22
Graphing Asymptotes in a Nutshell

49m 7s

Intro
0:00
Introduction
0:05
A Process for Graphing
1:22
1. Factor Numerator and Denominator
1:50
2. Find Domain
2:53
3. Simplifying the Function
3:59
4. Find Vertical Asymptotes
4:59
5. Find Horizontal/Slant Asymptotes
5:24
6. Find Intercepts
7:35
7. Draw Graph (Find Points as Necessary)
9:21
Draw Graph Example
11:21
Vertical Asymptote
11:41
Horizontal Asymptote
11:50
Other Graphing
12:16
Test Intervals
15:08
Example 1
17:57
Example 2
23:01
Example 3
29:02
Example 4
33:37
Partial Fractions

44m 56s

Intro
0:00
Introduction: Idea
0:04
Introduction: Prerequisites and Uses
1:57
Proper vs. Improper Polynomial Fractions
3:11
Possible Things in the Denominator
4:38
Linear Factors
6:16
Example of Linear Factors
7:03
Multiple Linear Factors
7:48
Irreducible Quadratic Factors
8:25
Example of Quadratic Factors
9:26
Multiple Quadratic Factors
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
V. Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

Intro
0:00
Introduction
0:05
Fundamental Idea
1:46
Expanding the Idea
2:28
Multiplication of the Same Base
2:40
Exponents acting on Exponents
3:45
Different Bases with the Same Exponent
4:31
To the Zero
5:35
To the First
5:45
Fundamental Rule with the Zero Power
6:35
To the Negative
7:45
Any Number to a Negative Power
8:14
A Fraction to a Negative Power
9:58
Division with Exponential Terms
10:41
To the Fraction
11:33
Square Root
11:58
Any Root
12:59
Summary of Rules
14:38
To the Irrational
17:21
Example 1
20:34
Example 2
23:42
Example 3
27:44
Example 4
31:44
Example 5
33:15
Exponential Functions

47m 4s

Intro
0:00
Introduction
0:05
Definition of an Exponential Function
0:48
Definition of the Base
1:02
Restrictions on the Base
1:16
Computing Exponential Functions
2:29
Harder Computations
3:10
When to Use a Calculator
3:21
Graphing Exponential Functions: a>1
6:02
Three Examples
6:13
What to Notice on the Graph
7:44
A Story
8:27
Story Diagram
9:15
Increasing Exponentials
11:29
Story Morals
14:40
Application: Compound Interest
15:15
Compounding Year after Year
16:01
Function for Compounding Interest
16:51
A Special Number: e
20:55
Expression for e
21:28
Where e stabilizes
21:55
Application: Continuously Compounded Interest
24:07
Equation for Continuous Compounding
24:22
Exponential Decay 0<a<1
25:50
Three Examples
26:11
Why they 'lose' value
26:54
Example 1
27:47
Example 2
33:11
Example 3
36:34
Example 4
41:28
Introduction to Logarithms

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
Thinking about Logs like Inverses
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(log)
5:53
Logarithm of a Power
7:44
Logarithm of a Product
10:07
Logarithm of a Quotient
13:48
Caution! There Is No Rule for loga(M+N)
16:12
Summary of Properties
17:42
Change of Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
4:11
Inverse Property
6:09
Solving by Inverses
7:25
Dealing with Equations
7:50
Example of Taking an Exponent or Logarithm of an Equation
9:07
A Useful Property
11:57
Bring Down Exponents
12:01
Try to Simplify
13:20
Extraneous Solutions
13:45
Example 1
16:37
Example 2
19:39
Example 3
21:37
Example 4
26:45
Example 5
29:37
Application of Exponential and Logarithmic Functions

48m 46s

Intro
0:00
Introduction
0:06
Applications of Exponential Functions
1:07
A Secret!
2:17
Natural Exponential Growth Model
3:07
Figure out r
3:34
A Secret!--Why Does It Work?
4:44
e to the r Morphs
4:57
Example
5:06
Applications of Logarithmic Functions
8:32
Examples
8:43
What Logarithms are Useful For
9:53
Example 1
11:29
Example 2
15:30
Example 3
26:22
Example 4
32:05
Example 5
39:19
VI. Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
Radians
2:08
Circle is 2 Pi Radians
2:31
One Radian
2:52
Half-Circle and Right Angle
4:00
Converting Between Degrees and Radians
6:24
Formulas for Degrees and Radians
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
Adjacent, Opposite, Hypotenuse
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
Other Quadrants
9:43
Mnemonic: All Students Take Calculus
10:13
Example 1: Convert, Quadrant, Sine/Cosine
13:11
Example 2: Convert, Quadrant, Sine/Cosine
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-1
Extra Example 2: All Angles and Quadrants
-2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

Intro
0:00
Amplitude and Period of a Sine Wave
0:38
Sine Wave Graph
0:58
Amplitude: Distance from Middle to Peak
1:18
Peak: Distance from Peak to Peak
2:41
Phase Shift and Vertical Shift
4:13
Phase Shift: Distance Shifted Horizontally
4:16
Vertical Shift: Distance Shifted Vertically
6:48
Example 1: Amplitude/Period/Phase and Vertical Shift
8:04
Example 2: Amplitude/Period/Phase and Vertical Shift
17:39
Example 3: Find Sine Wave Given Attributes
25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

Intro
0:00
Tangent and Cotangent Definitions
0:21
Tangent Definition
0:25
Cotangent Definition
0:47
Master Formula: SOHCAHTOA
1:01
Mnemonic
1:16
Tangent and Cotangent Values
2:29
Remember Common Values of Sine and Cosine
2:46
90 Degrees Undefined
4:36
Slope and Menmonic: ASTC
5:47
Uses of Tangent
5:54
Example: Tangent of Angle is Slope
6:09
Sign of Tangent in Quadrants
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

Intro
0:00
Secant and Cosecant Definitions
0:17
Secant Definition
0:18
Cosecant Definition
0:33
Example 1: Graph Secant Function
0:48
Example 2: Values of Secant and Cosecant
6:49
Example 3: Odd, Even, or Neither
12:49
Extra Example 1: Graph of Cosecant Function
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

Intro
0:00
Arcsine Function
0:24
Restrictions between -1 and 1
0:43
Arcsine Notation
1:26
Arccosine Function
3:07
Restrictions between -1 and 1
3:36
Cosine Notation
3:53
Arctangent Function
4:30
Between -Pi/2 and Pi/2
4:44
Tangent Notation
5:02
Example 1: Domain/Range/Graph of Arcsine
5:45
Example 2: Arcsin/Arccos/Arctan Values
10:46
Example 3: Domain/Range/Graph of Arctangent
17:14
Extra Example 1: Domain/Range/Graph of Arccosine
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

Intro
0:00
Inverse Trigonometric Function Domains and Ranges
0:31
Arcsine
0:41
Arccosine
1:14
Arctangent
1:41
Example 1: Arcsines of Common Values
2:44
Example 2: Odd, Even, or Neither
5:57
Example 3: Arccosines of Common Values
12:24
Extra Example 1: Arctangents of Common Values
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
VII. Trigonometric Identities
Pythagorean Identity

19m 11s

Intro
0:00
Pythagorean Identity
0:17
Pythagorean Triangle
0:27
Pythagorean Identity
0:45
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity
1:14
Example 2: Find Angle Given Cosine and Quadrant
4:18
Example 3: Verify Trigonometric Identity
8:00
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro
0:00
Main Formulas
0:19
Companion to Pythagorean Identity
0:27
For Cotangents and Cosecants
0:52
How to Remember
0:58
Example 1: Prove the Identity
1:40
Example 2: Given Tan Find Sec
3:42
Example 3: Prove the Identity
7:45
Extra Example 1: Prove the Identity
-1
Extra Example 2: Given Sec Find Tan
-2
Addition and Subtraction Formulas

52m 52s

Intro
0:00
Addition and Subtraction Formulas
0:09
How to Remember
0:48
Cofunction Identities
1:31
How to Remember Graphically
1:44
Where to Use Cofunction Identities
2:52
Example 1: Derive the Formula for cos(A-B)
3:08
Example 2: Use Addition and Subtraction Formulas
16:03
Example 3: Use Addition and Subtraction Formulas to Prove Identity
25:11
Extra Example 1: Use cos(A-B) and Cofunction Identities
-1
Extra Example 2: Convert to Radians and use Formulas
-2
Double Angle Formulas

29m 5s

Intro
0:00
Main Formula
0:07
How to Remember from Addition Formula
0:18
Two Other Forms
1:35
Example 1: Find Sine and Cosine of Angle using Double Angle
3:16
Example 2: Prove Trigonometric Identity using Double Angle
9:37
Example 3: Use Addition and Subtraction Formulas
12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle
-1
Extra Example 2: Prove Trigonometric Identity using Double Angle
-2
Half-Angle Formulas

43m 55s

Intro
0:00
Main Formulas
0:09
Confusing Part
0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle
0:54
Example 2: Prove Trigonometric Identity using Half-Angle
11:51
Example 3: Prove the Half-Angle Formula for Tangents
18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle
-1
Extra Example 2: Prove Trigonometric Identity using Half-Angle
-2
VIII. Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

Intro
0:00
Master Formula for Right Angles
0:11
SOHCAHTOA
0:15
Only for Right Triangles
1:26
Example 1: Find All Angles in a Triangle
2:19
Example 2: Find Lengths of All Sides of Triangle
7:39
Example 3: Find All Angles in a Triangle
11:00
Extra Example 1: Find All Angles in a Triangle
-1
Extra Example 2: Find Lengths of All Sides of Triangle
-2
Law of Sines

56m 40s

Intro
0:00
Law of Sines Formula
0:18
SOHCAHTOA
0:27
Any Triangle
0:59
Graphical Representation
1:25
Solving Triangle Completely
2:37
When to Use Law of Sines
2:55
ASA, SAA, SSA, AAA
2:59
SAS, SSS for Law of Cosines
7:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
8:44
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:30
Example 3: How Many Triangles Satisfy Conditions, Solve Completely
28:32
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely
-2
Law of Cosines

49m 5s

Intro
0:00
Law of Cosines Formula
0:23
Graphical Representation
0:34
Relates Sides to Angles
1:00
Any Triangle
1:20
Generalization of Pythagorean Theorem
1:32
When to Use Law of Cosines
2:26
SAS, SSS
2:30
Heron's Formula
4:49
Semiperimeter S
5:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
5:53
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:19
Example 3: Find Area of a Triangle Given All Side Lengths
26:33
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: Length of Third Side and Area of Triangle
-2
Finding the Area of a Triangle

27m 37s

Intro
0:00
Master Right Triangle Formula and Law of Cosines
0:19
SOHCAHTOA
0:27
Law of Cosines
1:23
Heron's Formula
2:22
Semiperimeter S
2:37
Example 1: Area of Triangle with Two Sides and One Angle
3:12
Example 2: Area of Triangle with Three Sides
6:11
Example 3: Area of Triangle with Three Sides, No Heron's Formula
8:50
Extra Example 1: Area of Triangle with Two Sides and One Angle
-1
Extra Example 2: Area of Triangle with Two Sides and One Angle
-2
Word Problems and Applications of Trigonometry

34m 25s

Intro
0:00
Formulas to Remember
0:11
SOHCAHTOA
0:15
Law of Sines
0:55
Law of Cosines
1:48
Heron's Formula
2:46
Example 1: Telephone Pole Height
4:01
Example 2: Bridge Length
7:48
Example 3: Area of Triangular Field
14:20
Extra Example 1: Kite Height
-1
Extra Example 2: Roads to a Town
-2
IX. Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

Intro
0:00
Introduction
0:04
Graphs as Location of 'True'
1:49
All Locations that Make the Function True
2:25
Understand the Relationship Between Solutions and the Graph
3:43
Systems as Graphs
4:07
Equations as Lines
4:20
Intersection Point
5:19
Three Possibilities for Solutions
6:17
Independent
6:24
Inconsistent
6:36
Dependent
7:06
Solving by Substitution
8:37
Solve for One Variable
9:07
Substitute into the Second Equation
9:34
Solve for Both Variables
10:12
What If a System is Inconsistent or Dependent?
11:08
No Solutions
11:25
Infinite Solutions
12:30
Solving by Elimination
13:56
Example
14:22
Determining the Number of Solutions
16:30
Why Elimination Makes Sense
17:25
Solving by Graphing Calculator
19:59
Systems with More than Two Variables
23:22
Example 1
25:49
Example 2
30:22
Example 3
34:11
Example 4
38:55
Example 5
46:01
(Non-) Example 6
53:37
Systems of Linear Inequalities

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
Test Points for Shading
11:42
Example of Using a Point
12:41
Drawing Shading from the Point
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
22:09
Rearrange Equations
22:21
Graph
22:49
Critical Solution is at the Vertex of the Overlap
23:40
Try Each Vertice
24:35
Example 1
24:58
Example 2
28:57
Example 3
33:48
Example 4
43:10
Nonlinear Systems

41m 1s

Intro
0:00
Introduction
0:06
Substitution
1:12
Example
1:22
Elimination
3:46
Example
3:56
Elimination is Less Useful for Nonlinear Systems
4:56
Graphing
5:56
Using a Graphing Calculator
6:44
Number of Solutions
8:44
Systems of Nonlinear Inequalities
10:02
Graph Each Inequality
10:06
Dashed and/or Solid
10:18
Shade Appropriately
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
X. Vectors and Matrices
Vectors

1h 9m 31s

Intro
0:00
Introduction
0:10
Magnitude of the Force
0:22
Direction of the Force
0:48
Vector
0:52
Idea of a Vector
1:30
How Vectors are Denoted
2:00
Component Form
3:20
Angle Brackets and Parentheses
3:50
Magnitude/Length
4:26
Denoting the Magnitude of a Vector
5:16
Direction/Angle
7:52
Always Draw a Picture
8:50
Component Form from Magnitude & Angle
10:10
Scaling by Scalars
14:06
Unit Vectors
16:26
Combining Vectors - Algebraically
18:10
Combining Vectors - Geometrically
19:54
Resultant Vector
20:46
Alternate Component Form: i, j
21:16
The Zero Vector
23:18
Properties of Vectors
24:20
No Multiplication (Between Vectors)
28:30
Dot Product
29:40
Motion in a Medium
30:10
Fish in an Aquarium Example
31:38
More Than Two Dimensions
33:12
More Than Two Dimensions - Magnitude
34:18
Example 1
35:26
Example 2
38:10
Example 3
45:48
Example 4
50:40
Example 4, cont.
56:07
Example 5
1:01:32
Dot Product & Cross Product

35m 20s

Intro
0:00
Introduction
0:08
Dot Product - Definition
0:42
Dot Product Results in a Scalar, Not a Vector
2:10
Example in Two Dimensions
2:34
Angle and the Dot Product
2:58
The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors
2:59
Proof of Dot Product Formula
4:14
Won't Directly Help Us Better Understand Vectors
4:18
Dot Product - Geometric Interpretation
4:58
We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are
7:26
Dot Product - Perpendicular Vectors
8:24
If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other
8:54
Cross Product - Definition
11:08
Cross Product Only Works in Three Dimensions
11:09
Cross Product - A Mnemonic
12:16
The Determinant of a 3 x 3 Matrix and Standard Unit Vectors
12:17
Cross Product - Geometric Interpretations
14:30
The Right-Hand Rule
15:17
Cross Product - Geometric Interpretations Cont.
17:00
Example 1
18:40
Example 2
22:50
Example 3
24:04
Example 4
26:20
Bonus Round
29:18
Proof: Dot Product Formula
29:24
Proof: Dot Product Formula, cont.
30:38
Matrices

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
Talking About Specific Entries
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
Matrix Addition
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

Intro
0:00
Introduction
0:06
Not All Matrices Are Invertible
1:30
What Must a Matrix Have to Be Invertible?
2:08
Determinant
2:32
The Determinant is a Real Number Associated With a Square Matrix
2:38
If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
3:40
Determinant of a 2 x 2 Matrix
4:34
Think in Terms of Diagonals
5:12
Minors and Cofactors - Minors
6:24
Example
6:46
Minors and Cofactors - Cofactors
8:00
Cofactor is Closely Based on the Minor
8:01
Alternating Sign Pattern
9:04
Determinant of Larger Matrices
10:56
Example
13:00
Alternative Method for 3x3 Matrices
16:46
Not Recommended
16:48
Inverse of a 2 x 2 Matrix
19:02
Inverse of Larger Matrices
20:00
Using Inverse Matrices
21:06
When Multiplied Together, They Create the Identity Matrix
21:24
Example 1
23:45
Example 2
27:21
Example 3
32:49
Example 4
36:27
Finding the Inverse of Larger Matrices
41:59
General Inverse Method - Step 1
43:25
General Inverse Method - Step 2
43:27
General Inverse Method - Step 2, cont.
43:27
General Inverse Method - Step 3
45:15
Using Matrices to Solve Systems of Linear Equations

58m 34s

Intro
0:00
Introduction
0:12
Augmented Matrix
1:44
We Can Represent the Entire Linear System With an Augmented Matrix
1:50
Row Operations
3:22
Interchange the Locations of Two Rows
3:50
Multiply (or Divide) a Row by a Nonzero Number
3:58
Add (or Subtract) a Multiple of One Row to Another
4:12
Row Operations - Keep Notes!
5:50
Suggested Symbols
7:08
Gauss-Jordan Elimination - Idea
8:04
Gauss-Jordan Elimination - Idea, cont.
9:16
Reduced Row-Echelon Form
9:18
Gauss-Jordan Elimination - Method
11:36
Begin by Writing the System As An Augmented Matrix
11:38
Gauss-Jordan Elimination - Method, cont.
13:48
Cramer's Rule - 2 x 2 Matrices
17:08
Cramer's Rule - n x n Matrices
19:24
Solving with Inverse Matrices
21:10
Solving Inverse Matrices, cont.
25:28
The Mighty (Graphing) Calculator
26:38
Example 1
29:56
Example 2
33:56
Example 3
37:00
Example 3, cont.
45:04
Example 4
51:28
XI. Alternate Ways to Graph
Parametric Equations

53m 33s

Intro
0:00
Introduction
0:06
Definition
1:10
Plane Curve
1:24
The Key Idea
2:00
Graphing with Parametric Equations
2:52
Same Graph, Different Equations
5:04
How Is That Possible?
5:36
Same Graph, Different Equations, cont.
5:42
Here's Another to Consider
7:56
Same Plane Curve, But Still Different
8:10
A Metaphor for Parametric Equations
9:36
Think of Parametric Equations As a Way to Describe the Motion of An Object
9:38
Graph Shows Where It Went, But Not Speed
10:32
Eliminating Parameters
12:14
Rectangular Equation
12:16
Caution
13:52
Creating Parametric Equations
14:30
Interesting Graphs
16:38
Graphing Calculators, Yay!
19:18
Example 1
22:36
Example 2
28:26
Example 3
37:36
Example 4
41:00
Projectile Motion
44:26
Example 5
47:00
Polar Coordinates

48m 7s

Intro
0:00
Introduction
0:04
Polar Coordinates Give Us a Way To Describe the Location of a Point
0:26
Polar Equations and Functions
0:50
Plotting Points with Polar Coordinates
1:06
The Distance of the Point from the Origin
1:09
The Angle of the Point
1:33
Give Points as the Ordered Pair (r,θ)
2:03
Visualizing Plotting in Polar Coordinates
2:32
First Way We Can Plot
2:39
Second Way We Can Plot
2:50
First, We'll Look at Visualizing r, Then θ
3:09
Rotate the Length Counter-Clockwise by θ
3:38
Alternatively, We Can Visualize θ, Then r
4:06
'Polar Graph Paper'
6:17
Horizontal and Vertical Tick Marks Are Not Useful for Polar
6:42
Use Concentric Circles to Helps Up See Distance From the Pole
7:08
Can Use Arc Sectors to See Angles
7:57
Multiple Ways to Name a Point
9:17
Examples
9:30
For Any Angle θ, We Can Make an Equivalent Angle
10:44
Negative Values for r
11:58
If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out
12:22
Another Way to Name the Same Point: Add π to θ and Make r Negative
13:44
Converting Between Rectangular and Polar
14:37
Rectangular Way to Name
14:43
Polar Way to Name
14:52
The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle
15:08
Connect Both Systems Through Basic Trigonometry
15:38
Equation to Convert From Polar to Rectangular Coordinate Systems
16:55
Equation to Convert From Rectangular to Polar Coordinate Systems
17:13
Converting to Rectangular is Easy
17:20
Converting to Polar is a Bit Trickier
17:21
Draw Pictures
18:55
Example 1
19:50
Example 2
25:17
Example 3
31:05
Example 4
35:56
Example 5
41:49
Polar Equations & Functions

38m 16s

Intro
0:00
Introduction
0:04
Equations and Functions
1:16
Independent Variable
1:21
Dependent Variable
1:30
Examples
1:46
Always Assume That θ Is In Radians
2:44
Graphing in Polar Coordinates
3:29
Graph is the Same Way We Graph 'Normal' Stuff
3:32
Example
3:52
Graphing in Polar - Example, Cont.
6:45
Tips for Graphing
9:23
Notice Patterns
10:19
Repetition
13:39
Graphing Equations of One Variable
14:39
Converting Coordinate Types
16:16
Use the Same Conversion Formulas From the Previous Lesson
16:23
Interesting Graphs
17:48
Example 1
18:03
Example 2
18:34
Graphing Calculators, Yay!
19:07
Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
19:11
Check Out the Appendix
19:26
Example 1
21:36
Example 2
28:13
Example 3
34:24
Example 4
35:52
XII. Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

Intro
0:00
Polar Coordinates
0:49
Rectangular Form
0:52
Polar Form
1:25
R and Theta
1:51
Polar Form Conversion
2:27
R and Theta
2:35
Optimal Values
4:05
Euler's Formula
4:25
Multiplying Two Complex Numbers in Polar Form
6:10
Multiply r's Together and Add Exponents
6:32
Example 1: Convert Rectangular to Polar Form
7:17
Example 2: Convert Polar to Rectangular Form
13:49
Example 3: Multiply Two Complex Numbers
17:28
Extra Example 1: Convert Between Rectangular and Polar Forms
-1
Extra Example 2: Simplify Expression to Polar Form
-2
DeMoivre's Theorem

57m 37s

Intro
0:00
Introduction to DeMoivre's Theorem
0:10
n nth Roots
3:06
DeMoivre's Theorem: Finding nth Roots
3:52
Relation to Unit Circle
6:29
One nth Root for Each Value of k
7:11
Example 1: Convert to Polar Form and Use DeMoivre's Theorem
8:24
Example 2: Find Complex Eighth Roots
15:27
Example 3: Find Complex Roots
27:49
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem
-1
Extra Example 2: Find Complex Fourth Roots
-2
XIII. Counting & Probability
Counting

31m 36s

Intro
0:00
Introduction
0:08
Combinatorics
0:56
Definition: Event
1:24
Example
1:50
Visualizing an Event
3:02
Branching line diagram
3:06
Addition Principle
3:40
Example
4:18
Multiplication Principle
5:42
Example
6:24
Pigeonhole Principle
8:06
Example
10:26
Draw Pictures
11:06
Example 1
12:02
Example 2
14:16
Example 3
17:34
Example 4
21:26
Example 5
25:14
Permutations & Combinations

44m 3s

Intro
0:00
Introduction
0:08
Permutation
0:42
Combination
1:10
Towards a Permutation Formula
2:38
How Many Ways Can We Arrange the Letters A, B, C, D, and E?
3:02
Towards a Permutation Formula, cont.
3:34
Factorial Notation
6:56
Symbol Is '!'
6:58
Examples
7:32
Permutation of n Objects
8:44
Permutation of r Objects out of n
9:04
What If We Have More Objects Than We Have Slots to Fit Them Into?
9:46
Permutation of r Objects Out of n, cont.
10:28
Distinguishable Permutations
14:46
What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?
14:48
Distinguishable Permutations, cont.
17:04
Combinations
19:04
Combinations, cont.
20:56
Example 1
23:10
Example 2
26:16
Example 3
28:28
Example 4
31:52
Example 5
33:58
Example 6
36:34
Probability

36m 58s

Intro
0:00
Introduction
0:06
Definition: Sample Space
1:18
Event = Something Happening
1:20
Sample Space
1:36
Probability of an Event
2:12
Let E Be An Event and S Be The Corresponding Sample Space
2:14
'Equally Likely' Is Important
3:52
Fair and Random
5:26
Interpreting Probability
6:34
How Can We Interpret This Value?
7:24
We Can Represent Probability As a Fraction, a Decimal, Or a Percentage
8:04
One of Multiple Events Occurring
9:52
Mutually Exclusive Events
10:38
What If The Events Are Not Mutually Exclusive?
12:20
Taking the Possibility of Overlap Into Account
13:24
An Event Not Occurring
17:14
Complement of E
17:22
Independent Events
19:36
Independent
19:48
Conditional Events
21:28
What Is The Events Are Not Independent Though?
21:30
Conditional Probability
22:16
Conditional Events, cont.
23:51
Example 1
25:27
Example 2
27:09
Example 3
28:57
Example 4
30:51
Example 5
34:15
XIV. Conic Sections
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
XV. Sequences, Series, & Induction
Introduction to Sequences

57m 45s

Intro
0:00
Introduction
0:06
Definition: Sequence
0:28
Infinite Sequence
2:08
Finite Sequence
2:22
Length
2:58
Formula for the nth Term
3:22
Defining a Sequence Recursively
5:54
Initial Term
7:58
Sequences and Patterns
10:40
First, Identify a Pattern
12:52
How to Get From One Term to the Next
17:38
Tips for Finding Patterns
19:52
More Tips for Finding Patterns
24:14
Even More Tips
26:50
Example 1
30:32
Example 2
34:54
Fibonacci Sequence
34:55
Example 3
38:40
Example 4
45:02
Example 5
49:26
Example 6
51:54
Introduction to Series

40m 27s

Intro
0:00
Introduction
0:06
Definition: Series
1:20
Why We Need Notation
2:48
Simga Notation (AKA Summation Notation)
4:44
Thing Being Summed
5:42
Index of Summation
6:21
Lower Limit of Summation
7:09
Upper Limit of Summation
7:23
Sigma Notation, Example
7:36
Sigma Notation for Infinite Series
9:08
How to Reindex
10:58
How to Reindex, Expanding
12:56
How to Reindex, Substitution
16:46
Properties of Sums
19:42
Example 1
23:46
Example 2
25:34
Example 3
27:12
Example 4
29:54
Example 5
32:06
Example 6
37:16
Arithmetic Sequences & Series

31m 36s

Intro
0:00
Introduction
0:05
Definition: Arithmetic Sequence
0:47
Common Difference
1:13
Two Examples
1:19
Form for the nth Term
2:14
Recursive Relation
2:33
Towards an Arithmetic Series Formula
5:12
Creating a General Formula
10:09
General Formula for Arithmetic Series
14:23
Example 1
15:46
Example 2
17:37
Example 3
22:21
Example 4
24:09
Example 5
27:14
Geometric Sequences & Series

39m 27s

Intro
0:00
Introduction
0:06
Definition
0:48
Form for the nth Term
2:42
Formula for Geometric Series
5:16
Infinite Geometric Series
11:48
Diverges
13:04
Converges
14:48
Formula for Infinite Geometric Series
16:32
Example 1
20:32
Example 2
22:02
Example 3
26:00
Example 4
30:48
Example 5
34:28
Mathematical Induction

49m 53s

Intro
0:00
Introduction
0:06
Belief Vs. Proof
1:22
A Metaphor for Induction
6:14
The Principle of Mathematical Induction
11:38
Base Case
13:24
Inductive Step
13:30
Inductive Hypothesis
13:52
A Remark on Statements
14:18
Using Mathematical Induction
16:58
Working Example
19:58
Finding Patterns
28:46
Example 1
30:17
Example 2
37:50
Example 3
42:38
The Binomial Theorem

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
Diagonal Addition of Terms
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
XVI. Preview of Calculus
Idea of a Limit

40m 22s

Intro
0:00
Introduction
0:05
Motivating Example
1:26
Fuzzy Notion of a Limit
3:38
Limit is the Vertical Location a Function is Headed Towards
3:44
Limit is What the Function Output is Going to Be
4:15
Limit Notation
4:33
Exploring Limits - 'Ordinary' Function
5:26
Test Out
5:27
Graphing, We See The Answer Is What We Would Expect
5:44
Exploring Limits - Piecewise Function
6:45
If We Modify the Function a Bit
6:49
Exploring Limits - A Visual Conception
10:08
Definition of a Limit
12:07
If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.
12:09
We Are Not Concerned with f(x) at x=c
12:49
We Are Considering x Approaching From All Directions, Not Just One Side
13:10
Limits Do Not Always Exist
15:47
Finding Limits
19:49
Graphs
19:52
Tables
21:48
Precise Methods
24:53
Example 1
26:06
Example 2
27:39
Example 3
30:51
Example 4
33:11
Example 5
37:07
Formal Definition of a Limit

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
This Game is About Limits
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

32m 40s

Intro
0:00
Introduction
0:08
Method - 'Normal' Functions
2:04
The Easiest Limits to Find
2:06
It Does Not 'Break'
2:18
It Is Not Piecewise
2:26
Method - 'Normal' Functions, Example
3:38
Method - 'Normal' Functions, cont.
4:54
The Functions We're Used to Working With Go Where We Expect Them To Go
5:22
A Limit is About Figuring Out Where a Function is 'Headed'
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & One-Sided Limits

32m 43s

Intro
0:00
Introduction
0:06
Motivating Example
0:56
Continuity - Idea
2:14
Continuous Function
2:18
All Parts of Function Are Connected
2:28
Function's Graph Can Be Drawn Without Lifting Pencil
2:36
There Are No Breaks or Holes in Graph
2:56
Continuity - Idea, cont.
3:38
We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'
3:52
Continuity - Definition
5:16
A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does
5:18
Discontinuous
6:02
Discontinuity
6:10
Continuity and 'Normal' Functions
6:48
Return of the Motivating Example
8:14
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided Limits
12:08
Limits of Piecewise Functions
14:12
'Breakover' Points
14:22
We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other
15:34
Example 1
16:40
Example 2
18:54
Example 3
22:00
Example 4
26:36
Limits at Infinity & Limits of Sequences

32m 49s

Intro
0:00
Introduction
0:06
Definition: Limit of a Function at Infinity
1:44
A Limit at Infinity Works Very Similarly to How a Normal Limit Works
2:38
Evaluating Limits at Infinity
4:08
Rational Functions
4:17
Examples
4:30
For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
5:22
There are Three Possibilities
6:36
Evaluating Limits at Infinity, cont.
8:08
Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
10:06
Two Good Ways to Think About This
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

Intro
0:00
Introduction
0:08
The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
0:16
Instantaneous Slop
0:22
Instantaneous Rate of Change
0:28
Slope
1:24
The Vertical Change Divided by the Horizontal
1:40
Idea of Instantaneous Slope
2:10
What If We Wanted to Apply the Idea of Slope to a Non-Line?
2:14
Tangent to a Circle
3:52
What is the Tangent Line for a Circle?
4:42
Tangent to a Curve
5:20
Towards a Derivative - Average Slope
6:36
Towards a Derivative - Average Slope, cont.
8:20
An Approximation
11:24
Towards a Derivative - General Form
13:18
Towards a Derivative - General Form, cont.
16:46
An h Grows Smaller, Our Slope Approximation Becomes Better
18:44
Towards a Derivative - Limits!
20:04
Towards a Derivative - Limits!, cont.
22:08
We Want to Show the Slope at x=1
22:34
Towards a Derivative - Checking Our Slope
23:12
Definition of the Derivative
23:54
Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point
23:58
Differentiation
24:54
Notation for the Derivative
25:58
The Derivative is a Very Important Idea In Calculus
26:04
The Important Idea
27:34
Why Did We Learn the Formal Definition to Find a Derivative?
28:18
Example 1
30:50
Example 2
36:06
Example 3
40:24
The Power Rule
44:16
Makes It Easier to Find the Derivative of a Function
44:24
Examples
45:04
n Is Any Constant Number
45:46
Example 4
46:26
Area Under a Curve (Integrals)

45m 26s

Intro
0:00
Introduction
0:06
Integral
0:12
Idea of Area Under a Curve
1:18
Approximation by Rectangles
2:12
The Easiest Way to Find Area is With a Rectangle
2:18
Various Methods for Choosing Rectangles
4:30
Rectangle Method - Left-Most Point
5:12
The Left-Most Point
5:16
Rectangle Method - Right-Most Point
5:58
The Right-Most Point
6:00
Rectangle Method - Mid-Point
6:42
Horizontal Mid-Point
6:48
Rectangle Method - Maximum (Upper Sum)
7:34
Maximum Height
7:40
Rectangle Method - Minimum
8:54
Minimum Height
9:02
Evaluating the Area Approximation
10:08
Split the Interval Into n Sub-Intervals
10:30
More Rectangles, Better Approximation
12:14
The More We Us , the Better Our Approximation Becomes
12:16
Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity
12:44
Finding Area with a Limit
13:08
If This Limit Exists, It Is Called the Integral From a to b
14:08
The Process of Finding Integrals is Called Integration
14:22
The Big Reveal
14:40
The Integral is Based on the Antiderivative
14:46
The Big Reveal - Wait, Why?
16:28
The Rate of Change for the Area is Based on the Height of the Function
16:50
Height is the Derivative of Area, So Area is Based on the Antiderivative of Height
17:50
Example 1
19:06
Example 2
22:48
Example 3
29:06
Example 3, cont.
35:14
Example 4
40:14
XVII. Appendix: Graphing Calculators
Buying a Graphing Calculator

10m 41s

Intro
0:00
Should You Buy?
0:06
Should I Get a Graphing Utility?
0:20
Free Graphing Utilities - Web Based
0:38
Personal Favorite: Desmos
0:58
Free Graphing Utilities - Offline Programs
1:18
GeoGebra
1:31
Microsoft Mathematics
1:50
Grapher
2:18
Other Graphing Utilities - Tablet/Phone
2:48
Should You Buy a Graphing Calculator?
3:22
The Only Real Downside
4:10
Deciding on Buying
4:20
If You Plan on Continuing in Math and/or Science
4:26
If Money is Not Particularly Tight for You
4:32
If You Don't Plan to Continue in Math and Science
5:02
If You Do Plan to Continue and Money Is Tight
5:28
Which to Buy
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
Do Your Research
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
Tips for Purchasing
9:17
Buy Online
9:19
Buy Used
9:35
Ask Around
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
Read the Manual
0:06
Skim It
0:20
Play Around and Experiment
0:34
Syntax
0:40
Definition of Syntax in English and Math
0:46
Pay Careful Attention to Your Syntax When Working With a Calculator
2:08
Make Sure You Use Parentheses to Indicate the Proper Order of Operations
2:16
Think About the Results
3:54
Settings
4:58
You'll Almost Never Need to Change the Settings on Your Calculator
5:00
Tell Calculator In Settings Whether the Angles Are In Radians or Degrees
5:26
Graphing Mode
6:32
Error Messages
7:10
Don't Panic
7:11
Internet Search
7:32
So Many Things
8:14
More Powerful Than You Realize
8:18
Other Things Your Graphing Calculator Can Do
8:24
Playing Around
9:16
Graphing Functions, Window Settings, & Table of Values

10m 38s

Intro
0:00
Graphing Functions
0:18
Graphing Calculator Expects the Variable to Be x
0:28
Syntax
0:58
The Syntax We Choose Will Affect How the Function Graphs
1:00
Use Parentheses
1:26
The Viewing Window
2:00
One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
2:01
For Example
2:30
The Viewing Window, cont.
2:36
Window Settings
3:24
Manually Choose Window Settings
4:20
x Min
4:40
x Max
4:42
y Min
4:44
y Max
4:46
Changing the x Scale or y Scale
5:08
Window Settings, cont.
5:44
Table of Values
7:38
Allows You to Quickly Churn Out Values for Various Inputs
7:42
For example
7:44
Changing the Independent Variable From 'Automatic' to 'Ask'
8:50
Finding Points of Interest

9m 45s

Intro
0:00
Points of Interest
0:06
Interesting Points on the Graph
0:11
Roots/Zeros (Zero)
0:18
Relative Minimums (Min)
0:26
Relative Maximums (Max)
0:32
Intersections (Intersection)
0:38
Finding Points of Interest - Process
1:48
Graph the Function
1:49
Adjust Viewing Window
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
Advanced Technique: Arbitrary Solving
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
Advanced Technique: Calculus
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28
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Lecture Comments (12)

1 answer

Last reply by: Professor Selhorst-Jones
Sat Feb 11, 2017 8:14 PM

Post by John Lins on February 11, 2017

Please, help me with this question:
Find Laplace Transform of the following signal:
x2 (t)=(1-(1-t)e-3t)u(t)

1 answer

Last reply by: John Lins
Sat Feb 11, 2017 8:06 PM

Post by John Lins on February 11, 2017

Hello professor vincent. Could you please help me to find this Laplace Transform?
x2 (t)=(1-(1-t)e-3t)u(t)

1 answer

Last reply by: Professor Selhorst-Jones
Fri Mar 25, 2016 5:24 PM

Post by Ru Chigoba on November 18, 2015

Hi I need help with this problem :

Find the inverse of each relation or function, and then determine if the inverse is a function.
1 f={1,3), (1,-1), (1,-3), (1,1)}   f-1=

1 answer

Last reply by: Professor Selhorst-Jones
Tue Mar 17, 2015 11:24 PM

Post by thelma clarke on March 17, 2015

is there no easier way to solve this it seem very confusing

1 answer

Last reply by: Professor Selhorst-Jones
Mon Oct 20, 2014 11:27 AM

Post by Saadman Elman on October 18, 2014

It was a great clarification. Thanks,
However, Inverse function is not only has to do with horizontal line test passing but also has to do with vertical test passing. You only stressed on Horizontal test passing. My professor stressed both.

1 answer

Last reply by: Professor Selhorst-Jones
Mon Jun 16, 2014 9:18 PM

Post by Joshua Jacob on June 15, 2014

Sorry if this is slightly vague but I'm a little but confused on the last example. Could you explain it in other words please?

Inverse Functions

  • A function does a transformation on an input. But what if there was some way to reverse that transformation? This is the idea of an inverse function: a way to reverse a transformation and get back to our original input.
  • To help us understand this idea, imagine a factory where if you give them a pile of parts, they'll make you a car. Now imagine another factory just down the road, where if you give them that car, they'll give you back the original pile of parts you started with. There is one process, but there is also an inverse process that gets you back to where you started. If you follow one process with the other, nothing happens.
  • It's important to note that not all functions have inverses. Some types of transformation cannot be undone. If the information about what we started with is permanently destroyed by the transformation, it cannot be reversed.
  • A function has an inverse if the function is one-to-one: for any a, b in the domain of f where a ≠ b, then f(a) ≠ f(b). Different inputs produce different outputs.
  • We can see this property in the graph of a function with the Horizontal Line Test. If a function is one-to-one, it is impossible to draw a horizontal line somewhere such that it will intersect the graph twice (or more).
  • Given some function f that is one-to-one, there exists an inverse function, f−1, such that for all x in the domain of f,
    f−1
    f(x)
    = x.
    In other words, when f−1 operates on the output of f, it gives the original input that went into f. [Caution: f−1 means the inverse of f, not [1/f]. In general, f−1 ≠ [1/f].]
  • We can figure out the domain and range for f−1 by looking at f. Since the set of all outputs is the range of f, and f−1 can take any output of f, the domain of f−1 is the range of f. Likewise, f−1 can output all possible inputs for f, so the range of f−1 is the domain of f.
  • The inverse of f−1 is simply f. This makes intuitive sense: if you do the opposite of an opposite, you end up doing the original thing.
  • Visually, f−1 is the mirror of f over y=x. This is because f−1 swaps the outputs and inputs from f, which is the same thing as swapping x and y by mirroring over y=x.
  • To find the inverse to a function, we effectively need a way to "reverse" the function. This can be a little bit confusing at first, so here is a step-by-step guide for finding inverse functions.

      1. Check function is one-to-one;    f(x) = x3+1
      2. Swap f(x) for y;     y = x3+1
      3. Interchange x and y;     x = y3+1
      4. Solve for y;     y = 3√{x−1}
      5. Replace y with f−1(x);     f−1(x) = 3√{x−1}
  • While this method will produce the inverse if followed correctly, it is not perfect. Notice that in steps #2 and #3 above, the equations are completely different, yet they still use the same x and y. Technically, it is not possible for x and y to fulfill both of these equations at the same time. What's really happening is that when we swap in #3, we're actually creating a new, different y. The first one stands in for f(x), but the second stands in for f−1(x). This implicit difference between y's can be confusing, so be careful. Make a note on your paper where you swap x and y so you can see the switch to "inverse world".
  • Taking inverses can be difficult: it's easy to make a mistake. This means it's important to check your work. By definition, f−1 ( f(x) ) = x. This means if you know what f−1(x) and f(x) are, you can just compose them! If it really is the inverse, you'll get x. Furthermore, since we know f( f−1 (x) ) = x as well, you can compose them in either order when checking.

Inverse Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Introduction 0:04
  • Analogy by picture 1:10
    • How to Denote the inverse
    • What Comes out of the Inverse
  • Requirement for Reversing 2:02
    • The Basketball Factory
    • The Importance of Information
  • One-to-One 4:04
    • Requirement for Reversibility
    • When a Function has an Inverse
    • One-to-One
    • Not One-to-One
    • Not a Function
  • Horizontal Line Test 7:01
    • How to the test Works
    • One-to-One
    • Not One-to-One
  • Definition: Inverse Function 9:12
    • Formal Definition
    • Caution to Students
  • Domain and Range 11:12
    • Finding the Range of the Function Inverse
    • Finding the Domain of the Function Inverse
  • Inverse of an Inverse 13:09
    • Its just x!
    • Proof
  • Graphical Interpretation 17:07
    • Horizontal Line Test
    • Graph of the Inverse
    • Swapping Inputs and Outputs to Draw Inverses
  • How to Find the Inverse 21:03
    • What We Are Looking For
    • Reversing the Function
  • A Method to Find Inverses 22:33
    • Check Function is One-to-One
    • Swap f(x) for y
    • Interchange x and y
    • Solve for y
    • Replace y with the inverse
  • Some Comments 25:01
    • Keeping Step 2 and 3 Straight
    • Switching to Inverse
  • Checking Inverses 28:52
    • How to Check an Inverse
    • Quick Example of How to Check
  • Example 1 31:48
  • Example 2 34:56
  • Example 3 39:29
  • Example 4 46:19

Transcription: Inverse Functions

Hi--welcome back to Educator.com.0000

Today, we are going to talk about inverse functions.0002

A function does a transformation on an input; we have talked about functions for a while now.0005

But what if there was some way to reverse that transformation?0009

This is the idea behind an inverse function: it is a way to reverse a transformation, reverse the process that another function is doing.0012

It is a way to get back to our original input.0021

By way of analogy, let's imagine a factory where, if you give them a pile of parts, they will make you a car.0024

Now, if you take that car down the road to this other factory, you can give them that car,0029

and they will give you back the original pile of parts you started with.0034

There is one factory where they make cars out of parts, but then there is a second factory0038

where they take cars and break them down into the original parts that were used to make them.0042

There is one process, but there is also an inverse process that gets you back to where you started.0046

If you follow one process with the other immediately, it ends up as if you haven't done anything.0051

If you bring the pile of parts to the first factory, and then take that car to the second factory,0055

and they give you back the pile of parts, it is like you just started with a pile of parts and didn't do anything to it.0059

This is the idea behind an inverse function: it reverses a process--it reverses a transformation and gets you back to where you started.0063

We have used the analogy of a function as a machine before;0071

and it is a good image for being able to get across what is going on with inverse functions, as well.0073

So, a function machine takes inputs, and it transforms them to outputs by some rule.0079

So, what we are used to is: we plug in x into the function f, and it gives out f acting on x, f(x).0084

Now, we could plug it into another one; we could plug it into an inverse machine, an inverse to f;0092

and that would be called "f inverse," the f with the little -1 in the corner.0097

f-1 denotes the inverse of f; we call it "f inverse."0103

We plug f(x) into f-1, into that machine; we get right back to our original x.0108

It is as if we hadn't done anything; the first machine does something, but then the second machine0114

reverses that process and gets us back to where we started.0119

So, is there a requirement for reversing--can we make an inverse out of all functions out there?0123

No; let's see why by analogy first.0129

Imagine a factory where, if you give them a pile of wood or a pile of metal, they give you a basketball in return.0133

The basketball is the exact same, whether you started with wood or metal; it is always the exact same basketball.0139

It doesn't matter what you gave them; it is just a basketball.0145

Now, let's say you walk down the road to another factory; you give them that basketball;0148

you tell them to reverse the process; and then you walk away--you give them no other information.0152

Can that factory take a basketball and transform it back into a pile of wood or a pile of metal, if all they have is a basketball and no other information?0156

No, they have no idea what you started with.0164

Maybe they have wood; maybe they have metal; but the point is that they have no way0168

to be able to know which one they are supposed to give you at this point.0172

They don't have the information; the only person who has the information is you, when you brought the original wood pile,0175

or brought the original metal pile--because all they have is the basketball, and the basketball could indicate wood, or it could indicate metal.0180

They have no way to know what you started with; there is no way to figure it out.0188

The information about what you originally had has been destroyed (although you would know it, because you brought it to the factory).0191

But assuming you forgot, then the information has been destroyed--no one has the information anymore.0197

Another way to think about it would be if you took a piece of paper, and you burned that piece of paper.0202

You would be left with a pile of ashes.0206

Now, someone could come along and think, "Oh, a pile of ashes--it used to be a piece of paper."0208

But if you take two pieces of paper, and you write two completely different things on the two pieces of paper,0212

and then you burn the two of them, a person could come along and think, "Two piles of ashes..."0217

And they would know it was paper, but they wouldn't know what was written on them.0221

They wouldn't be able to get that information back; the information has been destroyed.0224

They know it was paper, but they don't know what was written on the paper.0227

The basketball one...you have given them wood; you have given them metal; you get the same thing.0231

The information about what you started with has been lost; the information has been destroyed,0235

unless you come along and also say, "Oh, by the way, that basketball came from ____."0239

The issue, in this scenario, is that we have two inputs providing the same output--whether it is metal or wood, you get a basketball.0245

So, if we try to have a reverse on that, we have no way to know which one to go back to.0252

We don't know if we are going to go back to metal; we don't know if we are going to go back to wood,0256

because we don't know what the basketball is representing.0259

So, to be a reversible process--for it to be possible to reverse something--the process has to have a different output for every input.0262

If you give them metal, they have to give you one color of basketball; and if you give them wood, they give you a totally different color of basketball.0271

Then, the second factory would say, "Oh, that is a wood basketball" or "Oh, that is a metal basketball."0277

And they would be able to know what to do at that point.0281

So, for a function f to have an inverse, it has to be that, for any a and b in the domain, any a or b that we could use in f normally,0284

where a is not the same thing as b (where a and b are distinct from each other--they are different),0291

then f(a) is different than f(b); f(a) does not equal f(b).0297

So, if a and b are different, then the function's outputs on a and b are different, as well.0302

So, different inputs going into a function have to produce different outputs; we call this property one-to-one.0308

If this function has a property where whatever you put in, as long as it is different from something else going in,0316

it means the two things will be different, that is called one-to-one: different inputs produce different outputs.0323

You give them metal; you get one color of basketball; you give them wood--you get a different color of basketball.0328

Here are some examples, so we can see this in a diagram.0334

Here is an example that is one-to-one: a goes to 2; b goes to 1; c goes to 3.0336

They each go to different things: different inputs each have different places that they end up going.0344

Something that is not one-to-one: a goes to 1 and b goes to 1.0350

It doesn't matter that c goes to 3, because a and b have both gone to the same thing, so they have different inputs that are producing the exact same output.0354

a and b are different things, but they both produce a 1; so it is not one-to-one.0365

We have that copy; we are putting in wood, and putting in metal, and we get basketball in both cases.0369

And then, finally, just to remind us: this one over here (hopefully you remember this) is not a function.0373

And it is not a function, because b is able to produce two outputs at once; and that is something that is not allowed for a function.0378

If a function takes in one input, it is only allowed to produce one output; it can't produce multiple outputs from a single input.0384

So, why do we call it one-to-one--why are we using this word, "one-to-one"?0391

Well, we can think of it as being because a has one partner, and b has one partner, and c has one partner.0395

Everybody gets a partner, and nobody has shared partners; everybody gets their partner, and that partner is theirs.0402

They don't have to share it with anybody else.0408

It is one-to-one: this thing is matched to this thing, and there is nobody else who is going to match up to that one: one-to-one.0410

All right, how can we test for this?0417

One way to test for this, to test if a function is one-to-one: we know, if we are going to be one-to-one,0420

that every input must have a unique output; that was what it meant to be one-to-one.0425

If we have different inputs, we have different outputs.0429

So, if we draw a horizontal line on the graph, it can intersect the graph only once, or not at all.0432

Remember, if we have some picture on a graph--like if we have this point--then what that means0438

is that this is the input, and this over here is the output.0443

We make it a point: (input,output); that is why it is (x,f(x)).0449

If it is f(x) = x2, then we plug in 3, and we get (3,9), (3,32).0458

The input is our horizontal location; the output is our vertical location.0465

The horizontal line test is a way to test if the function has the same output for multiple inputs.0469

We draw a horizontal line across, because wherever an output hits the graph, we know that there must be an input directly below it.0477

If a function is not one-to-one, you will be able to draw a horizontal line that will intersect it twice, or maybe even more.0485

Let's look at some examples: first, here is one that is one-to-one, because whenever we draw0490

any sort of horizontal line, it is only going to cut it once.0496

The only place that might seem a little confusing is if we draw it near here.0499

It might make you think, "Well, doesn't it look like those are stacked?"0503

Well, remember, we can't draw perfectly what is being represented by the mathematics.0506

We have to give our lines some thickness; in reality, the line is infinitely thin.0511

So, while it looks like they are kind of getting stacked, it is actually still moving through that zone; it is not constant.0517

It is increasing just a little bit, but it isn't constant.0522

Let's look at one that is not one-to-one: over here, this horizontal line (or many horizontal lines that we could make)--0525

it cuts it in two places; so we know that, here and here, there are two inputs.0532

We can produce the same output from two different locations.0537

We have two inputs making one output; so that means we are not one-to-one,0541

because this one is partner to that height, but this one is also partner to this height.0545

So, we are not one-to-one, because we have to share an output.0550

Now that we have all of these ideas in mind, we are finally ready to define an inverse function; we can really talk about them and sink our teeth into them.0554

Given some function f that is one-to-one (it has to be one-to-one for this to happen),0561

there exists and inverse function, which we denote as f-1 such that, for all x in the domain of f,0566

any x that could normally go into f, any value that could normally be input,0574

f-1 acting on f(x) becomes just x.0578

So, we have f acting on x like normal; and then, f-1 acts on that whole thing.0583

And it breaks the action that was done by f and returns us back to our original input.0591

In other words, when f-1 operates on the output of f, it gives the original input that went into f.0596

Caution: I want to warn you about something: f-1 means the inverse of f, not 1/f.0604

This can be confusing, because, if you have taken algebra and remember your exponents0612

(you might have forgotten them, but we will talk about them later in this course), -1 can mean a reciprocal for numbers.0617

So, 7-1 becomes 1/7; x-1 becomes 1/x.0622

But f is not "to the -1"; it is just a symbol that says inverse--"This is a function inverse."0630

So, in general, for the most part, f-1, "f inverse," is not the same thing as 1/f.0636

The inverse of f is normally not the reciprocal of f, 1/f.0643

This exponent thing, where 7-1 is 1/7, is not the case for functions.0649

On a function, the -1 does not represent an exponent; it is not an exponent.0656

But it instead tells us function inverse; it is a way of saying, "This is an inverse function"; that is what it is telling us, not "flip it to the reciprocal."0661

How can we get domain and range for f-1?0673

We can figure these out by looking at f; remember, the set of all outputs from f is its range.0675

The things that x can get mapped to by f, what f is able to map x to, is the range of f.0687

The domain of f is everything that x can be, everything that we can plug into f.0695

And then, the range of f is everything that can come out of f.0701

Now, f-1 has to be able to take any output of f.0704

It is not very good at reversing if there are some numbers that it is not allowed to reverse.0709

So, it has to be able to reverse anything from f.0713

If it is able to reverse anything from f, then that means the range of f has to be everything that we can put into f-1.0716

So, the domain of f-1, the things we can put into f-1, is the range of f.0723

The domain of f-1 is the range of f.0728

Likewise, because f-1 then breaks that f(x) and turns it back into original inputs,0732

we must be able to turn it back into all of the original inputs, because all of the original outputs have to be over here.0738

So, anything that we can make it to, we have to be able to make it back from.0745

So, since we are able to get back everywhere, that means that we can output all the possible inputs for f.0748

Since we can output all of the possible inputs, because we can reverse any of the processes,0755

then it must mean that we are able to get the range of f-1 from the domain of f.0760

So, the domain of f tells us the range of f-1, what we are allowed to output with f-1.0767

And the domain of f-1 tells us the range of f.0772

So, the domain of f-1 is the things that f is able to output; and the range of f-1,0776

the things that f-1 is able to output, is what we can put into f, the domain of f.0784

This idea is going to let us prove something later on.0790

Now we can get to that proof--the inverse of an inverse: what is the inverse of an inverse?0793

In symbols, what is (f-1)-1--what do we do if we are going to take the inverse of something that is already doing inverses?0797

Now, it might seem a little surprising, but it turns out that the inverse of f-1 is just f.0806

The inverse of f-1 is f; it seems a little surprising, maybe, but it makes a sort of intuitive sense.0812

If you do the opposite of an opposite, you get to the original thing.0820

If you do an action, but then you are going to do the opposite of that action,0824

but then you do the opposite of the opposite of the action, then you must be back at your original action.0828

So, we might be able to believe this on an intuitive level; it makes sense, intuitively, that two opposites gets us back to where we started.0833

But let's see a proof of this fact, formally; let's see it in formal mathematics.0840

So, how do we get this started? Well, by definition, f-1 is the function where,0845

for any x, f-1 acting on f(x) is going to just give us our original x.0849

If f acts on an input, and then f-1 comes and acts on that, we get back to our original input.0856

Now, consider (f-1)-1: by this definition of the way inverses work, it must be that f inverse, inverse,0862

when it acts on the thing that it is an inverse of...f inverse, inverse, is an inverse of f inverse...0869

I know it is complicated to say...but this one right here is going to be the opposite action of f-1.0874

So, if we take any y (don't get too worried about x and y; remember, they are just placeholders for inputs),0881

similarly, for any y, (f-1)-1, acting on f-1(y),0887

is going to just get us right back to our original y.0892

It is the same structure as what is going on here with f-1(f(x)) = x: we are just reversing a process.0895

So, it doesn't matter that one of the processes is already a reversed process, because we are reversing this other reversed process.0901

So, we get back to our original thing.0907

Now, we know that y is in the domain of f-1, because we are allowed to put it into f-1.0910

It is allowed to go into f-1; now, we know, from our thing that we were just talking about,0917

that the domain of f-1 is the range of f; so there has to be...0922

If f-1 is the range of f (the domain of f-1 is the range of f),0930

if you are in the range, then that means that there is something out there that can produce this.0940

That means that, if you are in the range of f, there must be some x in the domain of f;0944

there has to be some way to get to that place in the range, so that f(x) is equal to y.0949

There is some x out there in the domain of our original f, that f(x) is equal to y.0954

So now, we have what we need: we can use this f(x) = y, and we can just plug it in right here and here.0960

We will plug it in for the two y's up here, and we will see what happens.0967

Thus, f inverse, inverse, acting on f-1(f(x))...0970

because remember, we know that there has to be some way to get an x such that f(x) = y,0974

because of this business about domain and range; so we plug that in here, and we plug that in here.0979

And we have that f inverse, inverse, on f inverse of f of x, must be equal to this over here on the right, as well.0985

We are just doing substitution.0993

But we know, by the definition of f-1(f(x)), that this just turns out to be x.0995

This whole thing right here just comes down to x--it simplifies right out to x.1002

So, it must be the case that f inverse, inverse, of x is the same thing as f(x).1007

If f(x) is the exact same thing as f inverse, inverse of x, it must be the case that (f-1)-1 is just the same thing as f.1012

And our proof is finished; great.1023

How can we interpret this graphically?--there is a great way to interpret inverses through graphs.1028

First, let's consider f(x) = x3 + 1.1032

Now, we know that this one has to be one-to-one, because it passes the horizontal line test.1035

We come along and try to cut this with any horizontal line; they are only going to be able to cut in one place.1040

Even here, where we have sort of seemed to flatten out, it is still moving, because we know it is x3 + 1.1045

And it never actually stops going up; it just slows down how fast it is going up.1051

And our lines have to have thickness, so while it kind of looks like they are stacked, they are not really.1055

So, we see that it passed the horizontal line test; so it must be one-to-one.1059

If it is one-to-one, we know it has to have an inverse; that is how we talked about this, right from the beginning.1064

Now, notice that the graph, any graph, is made up of the points (x,f(x)).1069

We talked about this before: 0 gets mapped to 1 when we plug it in as f(0) = 1; so that gives us the point (0,1).1073

That is how we make up our original graph for f.1082

Now, the graph of f-1 will swap these coordinates.1084

It takes in outputs and gives out inputs, in a way; so its input will swap these two things.1088

It takes in f(x), and it gives out x; so the points of f-1 will be the reverse of what we had for the points of the other one.1096

So, (f(x),x) is what we get for f-1.1105

Now, visually, what that means is that f-1 is going to be the mirror of y = x; and that is our line right here, y = x.1109

Why is that the case? Well, look: we swap x and y coordinates if we go across this,1117

because (-3,0) swaps to (0,-3); if we are going over y = x, if we are mirroring across this, we will swap the locations.1122

And so, if we mirror over y = x, we are going to swap x and y, y and x; we will swap the order of our points,1138

because y = x is sort of a way of saying, "Let's pretend for today that I am you and you are me."1145

y is going to pretend that it is x, and x is going to pretend that it is y.1153

They are sort of swapping places when we do a mirror over it.1156

So, that means our picture, mirroring f over y = x...we get the graph of f-1.1159

So, we look at this; we mirror over it; and we get places like this.1164

All right, we see how we are just sort of bouncing across it.1171

And this is going to happen with any of our inverses graphically.1179

So, any time f-1 is being looked at, we know it is going to be a bounce, a reflection through, a mirror over;1183

it is going to be symmetric to f with respect to the line y = x.1190

Since f-1 is swapping outputs and inputs, it is going to be sort of reversing the placements of these.1193

So, the graph of f-1 will always be symmetric to f, with respect to the line y = x.1201

It will bounce across, because when you bounce across y = x, you swap your coordinates.1206

Now, there are many ways to say this; I am saying "bounce across"; that is not really formal.1211

But we can formally say that it is symmetric to f, with respect to the line y = x.1215

You could also say that it reflects through y = x, or it reflects over y = x.1220

You could also say that it mirrors over, or it mirrors through, y = x.1224

There are many ways to say it; but in all of these things, the same idea is that we are going to bounce across,1227

and that that point will now show up at that same distance here.1232

So, let's see what it looks like: we bring them in, and indeed, they pop into those places.1235

They pop into being a nice, symmetric-to-the-line, y = x; and that makes sense.1240

We replaced the inputs with outputs and outputs with inputs; they have swapped locations.1247

We look at this one here, and the point (3,0) on the inverse is connected to (0,3) on the original function--the same sort of thing on both of them.1252

All right, so we have talked a lot about what is going on; we have a really great understanding of the mechanics behind an inverse.1264

But how do we actually find an inverse?1270

Now that we understand them, we are ready to actually go and find them.1273

How do we turn an algebraic function like f(x) = x3 + 1 into a formula?1276

Before we do this formula for f-1, consider that f-1 is taking the output of f(x); and it is transforming that into x.1282

To find a formula for f-1, we want a formula that gives x, if we know f(x).1290

Normally, f(x) = x3 + 1, for example--normally we have x.1298

We know x, and from that, we get our f(x); you plug in an x into a function, and it gives out f(x).1303

So, f-1 is the reverse of that; we know f(x), and we want to get x out of it.1312

So, to be able to do this, we are solving f(x) = x3 + 1 in reverse.1318

f(x) = x3 + 1; well, we move that over: f(x) - 1 = x3; so now we have 3√(f(x) - 1) = x.1323

If we know what f(x) is, we can figure out that that is what the original x that did it is.1338

We are solving it in reverse; we have reversed the function.1342

As opposed to solving f(x) in terms of x, we are solving x in terms of f(x).1345

Now, this is a little bit of a confusing idea; so instead, I am going to show you a method to do this.1351

The idea of reversing is really what is behind inverses; it can be a little hard to understand what to do on a step-by-step basis.1357

We are normally used to solving for f(x) in terms of x, having f(x) just on its own on one side, and having a bunch of stuff involving x on the other side.1365

So, at this point, it might be a little bit confusing for you to try to do it the other way.1372

And it would work; but let's learn a method that makes some of that confusion go away, and do things we are more used to doing.1376

Here is one step-by-step guide for finding inverse functions.1382

The very first thing we have to do: we have to check that the function is one-to-one.1384

It has to be one-to-one for us to be able to find an inverse at all.1388

Now, f(x) = x3 + 1...we just saw its graph; remember, it looked something like this.1391

So, we already know that it passes the horizontal line test; it does a great job; it is a great function.1397

It is one-to-one; great--we have already passed that part for this.1401

Next, we swap f(x) for y; this is going to be a little bit easier for us in solving.1406

We are used to solving for y's; we are not really used to solving for f(x)'s; so this will make it a little bit less confusing.1410

We switch out f(x) for y; great; in the next step, we interchange the x and the y.1415

In this one, we have x in its normal place and y in its normal place.1423

What we do on step 3 is swap their places: y takes the place of all of the x's, and x takes the place of y.1427

We swap x and y, interchange x and y; every time you had an x in step 2, you are now going to have a y;1436

every time you had a y (which is probably just the one time, since it was from a function), you are going to now have an x.1443

That is how we are doing this step that is the reversing step.1448

Solve for y: at this point, we have x = y3 + 1; so if x = y3 + 1, we solve for it.1453

We just move that 1 over: x - 1 = y3; and we have 3√(x - 1) = y.1460

And you will notice that this actually looks pretty much the exact same as what we just did on the previous slide--1469

but perhaps a little less confusing, because it is what we are used to seeing.1473

So, we have y = 3√(x - 1); and finally, just like we replaced f(x) with y, we now do a reverse replacement.1476

But we are now going to f-1; so y now becomes f-1(x).1486

f-1(x) is equal to 3√(x - 1); f-1(input) = 3√(input - 1).1492

Great; now, while this method will produce the inverse if followed correctly, it is not perfect.1500

Now, remember steps #2 and #3; in that, we had to swap f(x) for y, and then we were told to interchange x and y.1507

Remember, they swapped places; now notice, these equations are completely different.1515

They are totally, totally different from one another; yet they are still using the same x and y.1520

Technically, it is not possible to have both of these equations be true with the same x and y.1530

x and y can't possibly fulfill both of these equations at the same time, because they are completely different equations.1534

So, what is going on here? When we swap in step #3, we are really creating a new, different y.1541

When we have "swap f(x) for y," it is really red y or something here.1548

But then, when we do the interchange, it becomes a totally different color of y; it becomes like blue y here.1553

So, we are creating a new, different y; the y when we first swap is different than the meaning of the second y.1560

The swapping y is a different y from our first time that we replaced f(x).1567

The first one is standing in for f(x); that was our red y.1572

And then, the second one stands in for f-1(x); that is really taking the place there.1578

This implicit difference between y's can be confusing; so be careful.1584

I would recommend making a note on your paper; make a note when you are working that says where you swap.1588

Use a note to see that swap of x and y, so that you can see the switch over to this inverse world,1593

where you are now in an inverse world, and you can solve for an answer.1599

This is a bit confusing; so why are we learning this method, if it has this hidden, confusing1603

implicit difference, when we really think about what is going on?1609

In short, the reason we are doing this is because everybody else does.1612

That is not because it is perfect; it is because everyone else out there pretty much learns this method for solving inverses.1616

Most textbooks, and almost all of the teachers out there, teach this method.1623

So, it is important to learn, not because it is absolutely, perfectly correct, but because it is standard--1627

so that you can talk to other people, and talk about inverses, and they will understand1632

what you are talking about, because they are doing the same method that you are doing.1635

If you do something different, they might get confused.1639

If they are really clever, or they really understand what is going on, they will think, "Oh, yes, that makes perfect sense."1641

But we want to go with the standard method, so that other people will understand what we are doing.1645

And if we are taking a course at the same time as we are watching this course,1649

the teacher will think that is correct, as opposed to being confused by what you are doing and marking your grade down.1653

But the important idea here, the really important idea inside of this thing, that is confusing, is the reversal.1657

That is what the moment is all about--that moment between #2 and #3--the #3 step where we reverse, and we create this new y.1663

We reverse the places; instead of solving for an output, we are solving for input.1671

We are reversing the places, so we can do this directly; I did that with f(x), where I did f(x) = y;1678

and I solved directly for if we know what f(x) is.1685

I'm sorry, f(x) equals stuff involving x; I solved it directly for f(x)...1687

We had f(x) = x3 + 1, and we figured out that it also is the same thing that the cube root of f(x) - 1 is equal to x.1694

We figured that out; so there is this direct way of being able to do this.1706

We can do this directly; but lots of students find this difficult or confusing, so we have this method of swapping x and y.1709

And also, it has just become the standard way to do things; so it is good to practice this way, even though it is not absolutely perfect.1716

It is not a perfect method, but it does the job.1723

As long as you are careful and you pay attention to what you are doing--you closely follow its steps--1725

you will be able to get the answer, and you will be able to find the inverse function.1729

Taking inverses can be difficult; it seemed a little bit confusing from what I have been saying so far.1733

And it is an easy one to make a mistake on; this means it is really important to check your work.1737

You really want to make sure that you check your work on this.1743

How do you do this? Well, remember: by definition, f-1(f(x)) is equal to x.1746

That means, if we know what f-1 is (we have figured out its formula), and we know what f(x) is1752

(we were probably told f(x), we can just compose them.1756

We know how to compose them from our lesson Composite Functions.1760

If you didn't check out Composite Functions, you will have to watch that before you are able to compose them and do this check.1763

But if it is really the inverse, you will get x; if you compose f-1 with f(x), it has to come out to be x,1768

because that is the definition of how we are creating this stuff, right from the beginning.1775

Furthermore, we also know that f-1, inverse, was just f;1779

so it also must be the case that f acting on f-1(x) will give us x, as well.1785

You can compose them in either order when you are doing a check; and you will end up being able to get it correct.1790

Let's see a quick example: for example, if f(x) = x3 + 1, and f-1(x) = 3√(x - 1)1796

(the ones we have been working with), how do we check this?1803

Well, let's start with f-1(f(x)); we compose this: we plug in f(x) = x3 + 1.1805

So, f-1 acting on x3 + 1...now, remember, we are going to plug that into f-1(x).1814

But it is f-1(input); whatever is in the box just goes to the box over here.1821

So, it is going to be that f-1 will become cube root...where does the box go?1825

x3 + 1...that is our box...minus 1; so the cube root of x3 + 1 - 1...1832

+ 1 - 1 cancels; the cube root of x3 equals x; great--that checks out.1841

What about if we did it the other way--if we did it as f(f-1(x))?1847

Hopefully, this will work out, as well (and it will).1852

So, what is f-1(x)? f-1(x) is the cube root of x - 1, so f(3√(x - 1))...1854

what is going to happen over here?--we know that you plug in the box; you plug in the box.1864

So, f(3√(x - 1))...we are going to take that, and we are going to plug it in right here.1869

It is going to be 3√(x - 1), the quantity cubed, because it has to go in as the box; plus 1--finish out that function.1875

The cube root, cubed...those are going to cancel each other; we will get x - 1 + 1, which is just equal to x; and it checks out.1885

So, we can check it as f-1(f(x)) or f(f-1(x)); sometimes it might be easier for us to do it one way or the other.1895

We could also do both ways, if you want to check and be absolutely, doubly sure that we really got our work correct.1902

All right, let's move on to some examples.1907

Using these graphs for assistance, which of the following functions are one-to-one?1909

The first one is f(x) = 1/x; we do the horizontal line test--it is going to pass any high horizontal lines.1914

What about as we get lower? Well, we know that 1/x continues to move--it never freezes and becomes constant.1921

Does it ever cross this x-axis, though? No, it doesn't.1928

We haven't formally talked about asymptotes yet; we will talk about asymptotes in a later lesson.1932

But 1/x...as we go positive (f of a positive), 1 over a positive is going to also have to be positive.1936

So, it never crosses the x-axis; the same thing goes with the negatives--f of a negative is going to be a negative.1945

So, when it goes to the left, it never manages to cross this x-axis; as it goes to the right, it never manages to cross this x-axis.1951

And it keeps changing; so the two things never cross over each other.1958

So, yes, this is one-to-one.1961

What about the blue one, g(x) = x3 - 2x2 - x + 1?1968

It is easy to say it fails: we cross lots of places in the middle here, and it is able to have multiple points at the same time.1973

So, any one of these hits here and here and here; there are three points that all give the same output of 0; so it fails the horizontal line test.1982

It is not one-to-one.1992

Finally, (2x - 1) and (x2 + 1); 2x - 1 is just a line that is going to keep going on this way forever and ever and ever.1998

2x - 1, when x is less than or equal to 1...this is from piecewise functions; if you haven't checked out piecewise functions, this might be a little confusing.2007

But hopefully, you have watched that lesson already.2013

2x - 1 is x ≤ -1; it is just going to keep going on down and down and down, to the left and left and left.2015

And x2 + 1 is the right side of the parabola; if we plug in higher and higher numbers, it just keeps curving up and up and up to the right.2020

So, that means that we are never going to cross; the parabola is never going to double back and manage to touch itself again.2027

The parabola might eventually do this, but that part isn't on it.2033

And the line is never going to be able to go down to have itself crossed horizontally.2037

So, if we do any horizontal line crossing on this, it is never going to hit twice; so it is one-to-one.2042

One thing I would like to make a special comment on: notice that right here there is an empty space.2055

There is this gap where it jumps; is that a problem for a horizontal line test?2060

No, it is not a problem at all, because the horizontal line test is allowed to hit no points, as well.2064

It is allowed to hit one point or zero points; in this case, if it goes through that gap, it hits no points; but that is OK.2070

We are only worried about having multiple inputs for the same output.2076

It is OK if there are no inputs to make an output; the important thing2080

is that there are no double sets of inputs that all make the same output.2082

Like, in the blue one, where we had multiple different places where we could plug in some number--2087

plug in different numbers, but they would all produce zeroes.2092

All right, let's actually find an f-1: f(x) = -3x/(x + 3).2096

They told us, right from the beginning, that it is one-to-one; so we can jump right to figuring it out: what is f-1(x)?2102

And then, after it, we need to check our answer.2108

OK, so what is f-1(x)? Remember all of our steps, one by one.2111

f(x) = -3x/(x + 3): they told us, right from the beginning, that it is one-to-one, so we are already checked out.2115

We have already checked out the first one.2122

The next step: we swap y for f(x): y = -3x/(x + 3).2124

Now, that is not the important part of when we reverse, though; we reverse into inverse world.2131

So, here is when we go into inverse world; we reverse the place of x and y.2136

So now, it is x where y was, and it is -3y/(y + 3).2144

Now, we just go about this, and we solve it for y like we normally would.2151

Multiply both sides by y; we get x times (y + 3) equals -3y; let's distribute this out: xy + 3x...let's also move the 3y over, so + 3y = 0.2154

OK, at this point, we will pull out the y's from these two things; we will move them together, so we can see it a little bit easier at first.2170

xy + 3y + 3x = 0; let's subtract that 3x to move it over; -3x, -3x here.2176

So, then we will pull out the y's to the right; so we have x + 3, times quantity y, equals -3x.2186

Finally, we divide by that x + 3, and we get y = -3x/(x + 3).2195

And now, finally, we can plug in f-1 for this y; so we plug it in, and we get f-1(x) = -3x/(x + 3).2202

Great; now, let's check and make sure that we got this right.2216

We check this in red; here is our check--let's check it by plugging f into f-1.2219

So, we want this to come out to be x; it should be x, if we got everything right.2232

So, f-1(f(x)); what is f(x)? f(x) is this; and here is something funny to notice.2239

Notice -3x/(x + 3); amazingly, it just so happens that for -3x/(x + 3), f(x) and f-1(x) are the exact same thing--kind of impressive.2245

We plug this in; we have f-1(f(x)); f(x) is -3x/(x + 3); now, over here, we plug it in; what is in the box?2260

The box shows up here; the box shows up here; it shows up twice, so it is f-1 on -3x/(x + 3).2274

It is going to be -3...what is in the box?...-3x/(x + 3), over (-3x/(x + 3)) + 3.2281

Great; so the first thing that is going to be confusing is that we have this x + 3, and we have this x + 3 here.2301

So, let's take that out by multiplying the whole thing by (x + 3)/(x + 3).2307

We can get away with that, because it is just the same thing as 1: (x + 3)/(x + 3) is just 1.2312

So, (x + 3)/(x + 3)...multiply that here; the (x + 3) will cancel out here and cancel out here.2317

But remember, it also has to distribute to the other part, because they are not connected through multiplication on that; they are connected through addition.2324

So, we have -3, -3x, over -3x plus 3 times x plus 3.2330

These two negatives cancel out; so we have 3 times 3x on the top, -3x plus 3(x + 3)...so we have 9x on the top,2341

divided by -3x plus 3x plus 9; -3x plus 3x...they cancel each other out; we have 9x/9.2350

9 over 9...those cancel out, and we have just x.2362

So, that checks out--great, we have the answer.2367

All right, the next one: we have, this time, a piecewise function.2370

This is a little confusing: we didn't talk about this formula, but we will see how to do it.2375

f(x) = -x + 1 when x < 0, and -√x when x ≥ 0.2378

This is confusing; we don't know what to do about the different pieces of the piecewise function.2387

We don't know what to do about these two different categories: we have x < 0 and x ≥ 0.2393

We didn't learn that when we learned how to do inverses; but we could still figure out these two.2397

We could figure out what is the inverse of -x + 1 and what is the inverse of -√x.2402

We were told, explicitly, that this is one-to-one; so we can go ahead and do this, and then we will think about it.2407

First, we will do inverses on these two rules; and then we will figure out how they fit together--what are the categories for these two rules?2413

So, first, -x + 1; we will have y = -x + 1; we swap them, so we now get into our inverse world.2422

Swap their locations; we interchange them, and we are now at negative...sorry, not -x; the negative does not swap.2439

We are at x = -y + 1; we move the y over and move the x over; we get -x + 1; so y = -x + 1,2446

which is going to give us f-1 for at least the first rule here.2458

Now, what about the other one?--let's do that, as well.2464

So, y = -√x; we go into inverse mode; we reverse their locations; and we are now at x = -√y.2467

So, how do we solve for y? Well, we move this negative over: -x = √y.2482

Square both sides; we get (-x)2 = y; and then (-x)2...the negatives will cancel out, so we get just x2 = y.2487

And so, this is the inverse rule for this part.2498

Now, here is the part where we start thinking: we know that f-1 is going to break into a piecewise function using these two different things.2502

y = -x + 1...so it will be -x + 1 for the first rule, and then x2 for the second rule.2513

But the question is that we don't know what the categories are.2520

How do we figure out what the categories are?2524

Well, remember: if f goes from its domain to its range, let's call that a to b, then f-1 does the reverse of that.2526

f-1 goes from b to a; it does the reverse.2543

What that means is that the domain...the thing that determined which rule we used...we need to do the range to determine which way to get back.2548

The range on these two rules...now we are back to using f, so range on f...for -x + 1:2557

well, -x + 1 was x < 0; that was the category, so it has to be within those.2571

So, what can it go to? Well, if we plug in a really big negative number, like, say, -100, we will get -(-100) + 1; so we get 101.2577

So, as long as we keep plugging in more and more negative numbers, we get bigger and bigger numbers.2584

We are able to get all the way out to positive infinity, as we are really far in negative numbers.2587

What is the lowest that we can get to? Well, we could get really close to 1, as we plug in -0.00000001.2592

We are really close to being to 1, so we can get right up to 1; but we can't actually touch it.2601

We have to exclude it, so we use parentheses.2605

So, the range for the first rule is this: -x + 1 becomes this.2607

So, I will put a red dot on that, because that matches to this rule here.2614

Now, what about the range for the other rule?2619

The range on this rule is -√x; it has x ≥ 0 as its domain; what are the numbers we can get out of this?2622

What is the largest number we can get out of it?2630

The largest number we can get out of it is actually 0; why?--because, when we plug in any reasonably large positive number,2632

like, say, 100, then -√100 is -10; so as we get bigger and bigger positive numbers that we plug in,2639

we actually get more and more negative.2649

So, we can actually go to any negative number we want; we can go all the way down to negative infinity.2651

Can we actually reach 0? Yes, we actually can reach 0, because it is greater than or equal to.2655

So, if we plug in x = 0, we get -√0, which is just 0; and we put a bracket to indicate that we are actually allowed to do it.2660

This one is the range for -√x, that rule; it is going to get a green dot on it, because it matches to the green rule.2668

That means that -x + 1 is allowed to take in...what values? It is allowed to take in the range values.2677

It is allowed to do a reverse on anything that shows up in the range (1,∞).2684

Also notice: these two ranges, (1,∞) and (-∞,0], don't have any intersection.2689

They don't overlap at all, so we don't have any worries about pulling from one versus pulling from the other.2695

They will never get in each other's way.2700

So, for this one, -x + 1, if it is going to be allowed to go from 1 to infinity, then that means we can plug in anything into f-1,2703

where x is greater than 1, which is to say input; it is not the same x that was up here.2711

It is now just saying "placeholder--anything that we are plugging in."2719

What about x2? Well, that was the green dot--that was allowed to go from negative infinity up to 0.2722

So, it is allowed to have x ≤ 0; it is allowed to go all the way up to negative infinity, but it can only just get to touching 0.2727

It is allowed to actually have 0, though; x > 1 is not actually allowed to touch 1, but it is able to get as close as it wants.2735

And there is our piecewise inverse function.2741

It is a little bit difficult, but if you think about it, you do each of the inverses, and then you think about2747

"How do I get the domain for the inverse? I get it from the domain of f becoming the range of our inverse,2750

and the range of our f becoming the domain of our inverse."2760

So, what the original function was able to output to is what the inverse is allowed to take in.2766

And that is how we figured out these rules, these categories--what the categories were for these two different transformations.2773

All right, the final example: f and g are one-to-one functions; now, prove that f composed with g, inverse, is equal to g inverse composed with f inverse.2779

This might be a little daunting at first; these are weird symbols; we are not used to using these sorts of things.2791

So, if that is the case, let's remind ourselves: from composition, f composed with g, acting on x, is equal to f(g(x)).2795

Now, I said before: it makes things always, always, always easier to see it in that format.2806

What we want to show is that g-1 composed with f-1 (which would be g-1(f-1(x)))...2811

we want to show that this one here is an inverse to that one over there.2823

That is what we are trying to prove, that f composed with g inverse...2829

We know, by the definition of how this symbol works, by how inverses work...2833

f composed with g-1, acting on f composed with g, on x, is going to just leave us as if we had done nothing,2837

because we are putting an inverse on something.2845

So, we want to show that this means the exact same thing as this right here.2848

So, let's just try it out: we will set it up like this: f composed with g-1, acting on f composed with g, acting on x.2854

OK, so what does that become? Well, we know that f composed with g, acting on x, is the same thing as f(g(x)).2871

All right, what is f composed with g-1? Well, we know (from what we did over here)2881

that we can bring that into g-1 acting on f-1, acting on whatever is going into it.2887

What is going into it here is this whole thing; so, it is going to be g-1, acting on f-1, acting on f, acting on g, acting on x.2891

And then, we close up all of those parentheses.2908

That is a little bit confusing; but we are seeing inverses right next to functions: f-1 acting on f, acting on whatever is in there.2912

It just cancels out and gets us right back to what we originally had in there.2922

So, f-1 acting on f...that cancels out, and we get g-1, acting on whatever was in there, which was g(x).2926

So, g-1 acting on g(x)...the exact same thing: we get down to x; so we have proved it.2933

g-1 composed with f-1 is how we create f composed with g, inverse.2940

Great; we have proved it.2948

All right, I hope you have a much better idea of how inverses work at this point.2949

They can be a little bit confusing, but you have that method to be able to guide you through it.2952

Just follow it really carefully, step-by-step.2956

The danger is if you break from those steps and do something else; that is where you can make mistakes.2958

If you really understand what is going on, you don't even have to use that method.2963

But it really is the standard method, so it is a good idea to stick with it, just because it is what a lot of other people are used to using.2966

And you can find it in a lot of textbooks.2972

All right, we will see you at Educator.com later--goodbye!2974