  Vincent Selhorst-Jones

Composite Functions

Slide Duration:

Section 1: Introduction
Introduction to Precalculus

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
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
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
18:50
19:04
19:21
20:04
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
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
Section 2: 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
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
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
Section 3: 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
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
13:08
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
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
9:56
11:38
Derivation
11:43
Final Form
12:23
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
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52

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
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
26:49
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
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
19:24
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
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
7:41
8:55
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
Section 4: 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
8:25
9:26
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
Section 5: 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
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
Section 6: Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
2:08
2:31
2:52
Half-Circle and Right Angle
4:00
6:24
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
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
9:43
Mnemonic: All Students Take Calculus
10:13
13:11
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
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
Section 7: 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

52m 52s

Intro
0:00
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
Section 8: 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
Section 9: 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
11:42
Example of Using a Point
12:41
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
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
Section 10: 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
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
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
Section 11: 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
Section 12: 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
Section 13: 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
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
Section 14: 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
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
11:51
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
Section 15: 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
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
Section 16: 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
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
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
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
Section 17: Appendix: Graphing Calculators

10m 41s

Intro
0:00
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
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
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
9:17
9:19
9:35
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
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
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
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
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
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
Bookmark & Share Embed

## Copy & Paste this embed code into your website’s HTML

Please ensure that your website editor is in text mode when you paste the code.
(In Wordpress, the mode button is on the top right corner.)
×
• - Allow users to view the embedded video in full-size.
Since this lesson is not free, only the preview will appear on your website.

• ## Related Books 1 answer Last reply by: Professor Selhorst-JonesFri Mar 13, 2015 9:02 PMPost by Mohammed Fayaz Amanulla on March 13, 2015 Example 2 question 2, why do i see a 12x. isnt it just 2x^2+3^2-7 which equals to 4x^2+9-7? 1 answer Last reply by: Professor Selhorst-JonesSat Feb 7, 2015 3:35 PMPost by Randall Tew on February 7, 2015Hello Professor Selhourst-Jones. I have a question about the 4th step in green pencil. Specifically, what rule enables the cancellation of the ^3 power in 3^3sqrt with the denominator 3 in 4/3? 1 answer Last reply by: Professor Selhorst-JonesTue Jan 6, 2015 12:45 PMPost by Andrew Demidenko on January 4, 2015Could you please give in answer another word problem / thanks / AD 1 answerLast reply by: Hari KaranWed Mar 26, 2014 2:38 AMPost by Hari Karan on March 26, 2014sir, could i ask you, for example 2 question 2, why do i see a 12x. isnt it just 2x^2+3^2-7 which equals to 4x^2+9-7? 0.o 1 answer Last reply by: Professor Selhorst-JonesThu Oct 17, 2013 8:27 AMPost by Constance Kang on October 17, 2013i have a question about example 3. you said taht the domain is x greater than -1 because the root will break if its not a positive number. however, i rationalized the bottom and got (x+1+root(x+1))over 2x+2 at the bottom, which makes the domain any real number except -1 1 answer Last reply by: Professor Selhorst-JonesSun Jul 28, 2013 9:39 PMPost by Chudamuni Dahal on July 28, 2013So prof, Can we rewrite f/g(h(x)) as[f(h(x))]/[g(h(x))] when asked for domain and other time? 0 answersPost by Chudamuni Dahal on July 28, 2013Prof, I have a question. Do we have to divide f(x)/g(x) first then only compose? Can't we do like this? f/g(h(x))so that the Domain is going to be all real number such that X is greater than or equal to -1. I am confused over why are we divide f/g first?I am confused Because you said earlier, composition to the nearest first. Other than that, thank you so much. So really have amazing ability to explain the concept in a such simple way 1 answer Last reply by: Professor Selhorst-JonesThu Jul 11, 2013 1:09 PMPost by Sarawut Chaiyadech on July 1, 2013Thx:) 5 answers Last reply by: Professor Selhorst-JonesThu Aug 8, 2013 3:32 PMPost by Jorge Sardinas on May 5, 2013isn't supposed to be 30 root 30 instead of 3 root 30 mr.jones????!!!!!!

### Composite Functions

• Two (or more) functions can interact with each other through good old arithmetic: addition, subtraction, multiplication, and division. These sorts of interactions are called arithmetic combinations. Here are the four types:
• Sum: (f+g)(x) = f(x) + g(x)
• Difference: (f−g)(x) = f(x) − g(x)
• Product: (fg)(x) = f(x) ·g(x)
• Quotient: (f/g)(x) = f(x)/g(x)        [and it must be that g(x) ≠ 0]
• A much more interesting idea is to compose two functions. Instead of giving both functions the same input, we give the input to just one function. Then we take the first function's output, and plug that in to the second function. The second function is acting on the first function. [If this idea is confusing, make sure to watch the video where we see some analogies.]
• For function composition, we can use the notation of f °g. [Read as "f composed with g."] If f °g acts on x, we have (f°g ) (x). This means g acts on x first, then f acts on whatever results. [Notice how the functions act in order of closeness to the original input.]
• Another (much easier) way to see (f °g ) (x) is in the function notation format we're already used to:
 ⎛⎝ f °g ⎞⎠ (x) = f ⎛⎝ g(x) ⎞⎠ .
(Recommendation: If you see the ° notation [such as f °g], rewrite it in the "normal" format [such as f ( g(x) )]. This normally makes it easier to understand and solve problems.)
• Working with composite functions might seem intimidating at first, but it's really just about plugging in appropriately. Each function has its own "rule", so composing multiple functions just means using the rules in succession.
• This idea of plugging in is shown beautifully by the notation f ( g(x) ). The function g acts on x, then f acts on the resulting g(x). Since we almost certainly know what g(x) looks like from the problem, we just use that as input for f.

### Composite Functions

Let f(x) = x2+1 and g(x) = 3x−2. What is (f+g)(x)? What is (f+g)(3)?
• The function (f+g)(x) is a new function created by an arithmetic combination of f(x) and g(x). It is defined as
 (f+g)(x) = f(x) + g(x).
• To find (f+g)(x), just substitute into the above:
 (f+g)(x) = [x2 +1] + [3x−2].
• Once you know (f+g)(x), just plug into it to find (f+g)(3). [Alternatively, you can find f(3) and g(3), then add them together. But in this case, we already had to figure out (f+g)(x), so we might as well use it.]
(f+g)(x) = x2 + 3x −1;     (f+g)(3) = 17
Let f(x) = 2x2−2 and g(x) = 3x2−4. What is (f−g)(x)? What is (f−g)(−4)?
• The function (f−g)(x) is a new function created by an arithmetic combination of f(x) and g(x). It is defined as
 (f−g)(x) = f(x) − g(x).
• To find (f−g)(x), just substitute into the above:
 (f−g)(x) = [2x2 − 2] − [3x2−4].
• Once you know (f−g)(x), just plug into it to find (f−g)(−4). [Alternatively, you can find f(−4) and g(−4), then subtract one from the other. But in this case, we already had to figure out (f−g)(x), so we might as well use it.]
(f−g)(x) = −x2 + 2;     (f−g)(−4) = −14
Let f(x) = 3x and g(x) = −2x2+11. What is (fg)(x)? What is (fg)(2)?
• The function (fg)(x) is a new function created by an arithmetic combination of f(x) and g(x). It is defined as
 (fg)(x) = f(x) ·g(x).
• To find (fg)(x), just substitute into the above:
 (fg)(x) = [3x] ·[−2x2+11].
• Once you know (fg)(x), just plug into it to find (fg)(2). [Alternatively, you can find f(2) and g(2), then multiply them together. But in this case, we already had to figure out (fg)(x), so we might as well use it.]
(fg)(x) = −6x3 + 33x;     (fg)(2) = 18
Let f(x) = x2+1 and g(x) = 4x−8. What is ([f/g])(x)? What is ([f/g])(−5)? What is the domain of ([f/g]) (x)?
• The function ([f/g])(x) is a new function created by an arithmetic combination of f(x) and g(x). It is defined as
 ⎛⎝ f g ⎞⎠ (x) = f(x) g(x) .
• To find ([f/g])(x), just substitute into the above:
 ⎛⎝ f g ⎞⎠ (x) = [x2+1] [4x−8] .
• Once you know ([f/g])(x), just plug into it to find ([f/g])(−5). [Alternatively, you can find f(−5) and g(−5), then divide one by the other. But in this case, we already had to figure out (fg)(x), so we might as well use it.]
• To find the domain, look for any values that would "break" ([f/g])(x). In this case, the function will be broken only if the denominator equals 0. Thus, find the x-value such that 4x−8=0. This x-value is forbidden, but all others will work fine.
[f/g])(x) = [(x2+1)/(4x−8)];     ([f/g])(−5) = −[13/14];     Domain: x ≠ 2
Let f(x) = x2 and g(x) = x−3. Find f(g(x)) and g(f(x)).
• Plugging one function into another is called function composition. Using it as simple as substituting the plugged-in function for the variable that would normally be there.
• f(g(x)) = f(x−3) = (x−3)2
• g(f(x)) = g( x2 ) = (x2) − 3
f(g(x)) = x2 − 6x + 9;     g(f(x)) = x2 − 3
Let f(x) = 4x+3 and g(x) = 1−x. Find (f°g)(x) and (g°f)(x).
• The ° notation denotes function composition. It says that the function on the left acts on the output of the function on the right.
• We can rewrite the ° notation in the function notation we're already used to:
 ⎛⎝ f°g ⎞⎠ (x) = f ⎛⎝ g(x) ⎞⎠ and ⎛⎝ g°f ⎞⎠ (x) = g ⎛⎝ f(x) ⎞⎠
From this, it's as simple as substituting the plugged-in function for the variable that would normally be there.
• (f°g)(x) = f(1−x) = 4(1−x)+3
• (g°f)(x) = g( 4x+3 ) = 1−(4x+3)
(f°g)(x) = −4x+7;     (g°f)(x) = −4x−2
Let f(x) = |x+3| and g(x) = 2x−10. Find (f°g)(x) and (g°f)(x).
• The ° notation denotes function composition. It says that the function on the left acts on the output of the function on the right.
• We can rewrite the ° notation in the function notation we're already used to:
 ⎛⎝ f°g ⎞⎠ (x) = f ⎛⎝ g(x) ⎞⎠ and ⎛⎝ g°f ⎞⎠ (x) = g ⎛⎝ f(x) ⎞⎠
From this, it's as simple as substituting the plugged-in function for the variable that would normally be there.
• (f°g)(x) = f(2x−10) = |(2x−10) + 3|
• (g°f)(x) = g( |x+3| ) = 2·|x+3| −10
(f°g)(x) = |2x−7|;     (g°f)(x) = 2·|x+3| −10
Let f(x) = [1/x] and g(x) = x−5. Find (f°g)(x) along with its domain.
• The ° notation denotes function composition. It says that the function on the left acts on the output of the function on the right. NOTE: This is even more important in this problem than in previous ones. Because one function acts on the output of the other, that output must exist. We have to be careful about the domains of functions at every step.
• We can rewrite the ° notation in the function notation we're already used to:
 ⎛⎝ f°g ⎞⎠ (x) = f ⎛⎝ g(x) ⎞⎠
From this, it's as simple as substituting the plugged-in function for the variable that would normally be there. NOTE: For a given x to be in the domain of (f°g)(x), it must be that x is in the domain of g and that g(x) is in the domain of f.
• (f°g)(x) = f(x−5) = [1/(x−5)]
• Because (f°g)(x) = f(x−5) = [1/(x−5)], it "breaks" when the denominator equals 0. Thus the domain of (f°g)(x) is x ≠ 5. [We don't have to worry about g(x) affecting the domain because the domain of g(x) is all numbers.]
(f°g)(x) = [1/(x−5)];     Domain: x ≠ 5
Let f(x) = x2−4 and g(x) = √{x−4}. Find (f°g)(x) along with its domain.
• The ° notation denotes function composition. It says that the function on the left acts on the output of the function on the right. NOTE: This is even more important in this problem than in previous ones. Because one function acts on the output of the other, that output must exist. We have to be careful about the domains of functions at every step.
• We can rewrite the ° notation in the function notation we're already used to:
 ⎛⎝ f°g ⎞⎠ (x) = f ⎛⎝ g(x) ⎞⎠
From this, it's as simple as substituting the plugged-in function for the variable that would normally be there. NOTE: For a given x to be in the domain of (f°g)(x), it must be that x is in the domain of g and that g(x) is in the domain of f.
• (f°g)(x) = f(√{x−4}) = (√{x−4})2−4
• Because g(x) = √{x−4}, it "breaks" when there is a negative inside the root. Thus the domain of g(x) is x ≥ 4. This domain is passed up to (f°g)(x) as well, since f must act upon g(x). So even though (f°g)(x) has no square root once simplified, it is still restricted by that domain.
(f°g)(x) = x−8;     Domain: x ≥ 4
Let f(x) = 2x, g(x) = x2+4, and h(x) = 5x−20. What is (h °f °g °f) (x)?
• The ° notation denotes function composition. It says that the function to the left acts on the output of the function to the right.
• We can rewrite the ° notation in the function notation we're already used to:
 ⎛⎝ h °f °g °f ⎞⎠ (x) = h ⎛⎝ f ⎛⎝ g ⎛⎝ f(x) ⎞⎠ ⎞⎠ ⎞⎠
From this, it's just a matter of substituting the plugged-in function for the variable that would normally be there, then repeating the process until fully simplified.
• h( f (g (f(x) ) ) ) = h( f (g (2x) ) )
• h( f (g (2x) ) ) = h( f ([2x]2 + 4) ) ) = h( f (4x2 + 4) ) )
• h( f (4x2 + 4) ) ) = h( 2[4x2 +4] ) ) = h( 8x2 +8 )
• h( 8x2 +8 ) = 5 [ 8x2+8] − 20 = 40x2 +20
(h °f °g °f) (x) = 40x2 + 20

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Composite 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
• Arithmetic Combinations 0:40
• Basic Operations
• Definition of the Four Arithmetic Combinations
• 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
• Another Visual Interpretation 7:17
• How to Use Composite Functions 8:21
• Example of on Function acting on Another
• Example 1 11:03
• Example 2 15:27
• Example 3 21:11
• Example 4 27:06

### Transcription: Composite Functions

Hi--welcome back to Educator.com.0000

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

We are often going to have 2 or even more functions that interact with each other.0005

This lesson will explore the fundamental ways that functions can interact with each other.0009

First, we will look at how functions can interact through just arithmetic: addition, subtraction, multiplication, and division.0014

These sorts of interactions are called arithmetic combinations, because they are just using arithmetic.0019

Second, we will move on to a more complex idea, using one function's output as another function's input.0024

We call this idea composition of functions; if we want to talk about a specific example, we call that a composite function, when we put multiple functions together.0031

All right, let's go--let's say we have two functions, f and g, and f(x) = x2, and g(x) = x--nice, basic functions.0039

Now, it is easy to imagine creating a new function that just adds f and g together; we would call it f + g.0049

That is not very imaginative, but it makes sense.0055

It would give us the sum of the two functions: the new function, f + g (x), would be equal to x2 + x.0057

We are just adding the two functions together; so we know each function is x2 and x, so we just add them together.0066

That is a simple, basic idea; we are using a basic arithmetic operation, and we are just putting them together through that.0072

We have arithmetic; let's use it.0079

We could expand this idea to the other three basic operations.0081

We could do this with subtraction: f - g would become x2 - x; fg(x) would be x2 times x (fg being f times g,0084

just like when we say 3x, we mean 3 times x); and f/g would be x2/x--as simple as that.0093

Given two functions f and g, along with an x that is in the domain of both, we defined these four different arithmetic combinations.0101

If x means something for f(x), and x means something for g(x)--it doesn't fail, like if we had √x be one of them,0108

we couldn't plug in -3; but as long it is in the domain of both--it is a number that both of them can accept and work on--0115

all of these work really well; sum is f + g(x)--just break it down into adding the two together.0121

Difference is just subtracting one, just like normal subtraction.0127

Product--when we have fg, we read it as times: f times g; and quotient is f(x)/g(x).0132

And it also has to be that g(x) does not equal 0, because otherwise we could accidentally end up0141

blowing the world up when we divide by 0, since we are not allowed to divide by 0,0146

because it is nonsense and doesn't mean anything.0149

So, since you can't divide by 0, it is not going to be defined when g(x) is 0, since you would have to divide by 0.0151

But other than that, we are pretty good to go; if it means something, if it comes out as a normal output, it is defined;0157

that input is in the domain, and it is defined as an output; then we can just put the two together.0162

We just put what f(x) is together and put what g(x) is together with any arithmetic combination that we want to; great.0167

That is a nice, direct idea; an arithmetic combination makes sense.0174

We put in the same input to the two functions, and then we combine their outputs with some pre-decided arithmetic operation.0177

If it is sum, we do it through addition; if it is product we do it through multiplication; things like that.0182

But we can do something more interesting: we can compose one function with another.0188

Instead of giving both functions the same input, we give the input to one function.0193

Then we take the first function's output, and we plug that into the second function.0198

Input goes into one, and then an output comes out of that; and that immediately goes into the second function.0203

And then finally, we get an output of that; the second function is acting on the first function.0207

Many lessons back, we first introduced the idea of a function; and we talked about how we can view it as a machine.0213

It takes in inputs, and the function produces outputs: x goes into the machine, the function f;0219

and then it gets put out after having been acted on.0225

The function is some process; it does some transformation on x, so we get f(x), f having acted upon x.0228

All right, so that is the idea of it as a machine.0237

We can expand this idea into function composition.0240

Function composition is just linking multiple machines together in series; we just put multiple of them together.0243

The output of the first function goes directly into the second as its input.0248

Our first input goes into, say, g; and so, it is now g(x); and then we plug all of g(x) into f.0252

And so, we have all of g(x) now being acted upon by f; input into the first machine, and output comes out of that machine.0264

And then, we just jam that right into the second machine.0272

And if we wanted, we could string this up...3, 4, 5, 6, 7...we could string up as many of these machines in order as we wanted.0274

We could compose as many functions as we wanted to; but it is easy to start by thinking about it in terms of two functions being composed together.0280

We note the previous slide's composition, when it went into g first, and then went into f, as f composed with g.0287

This is just a little circle between them: f circle g...we read that as f composed with g.0295

If f composed with g acts on x, acts on some input x, we have f composed with g of x, just like we would normally.0301

We have created a new function out of putting the two together.0308

By linking those two machines together, it is effectively one larger machine that is doing a new way of working.0311

This means that in this machine, f composed with g, g will act on x first; and then f will act on whatever results.0318

So, we have f composed with g; and we can break it down into g goes first; then f goes on what results, the thing that comes out of that.0327

Now, notice: the functions act in order of closeness.0336

g goes on first; and then, f goes on second, because it is farther away.0340

We hit it with the things that are closest to the input that we are putting in.0348

So, g goes onto the x, and then f goes onto what results; and if we had even more stacked up,0352

whatever was even further to the left would act after that.0356

The functions act in order of closeness to the original input.0360

There is another, much easier, way to see f composed with g of x in the function notation format that we are already used to--0363

the thing that we have been using for quite a while now.0370

f composed with g of x is just f(g(x)); so f composed with g of x...remember how we broke it up:0372

this one went first; and then f acted second--well, that is what we have right here.0380

g is acting first, and then f is acting second--it makes a lot of sense.0387

I would personally recommend, any time you see this circle notation--this f composed with g stuff--rewrite it0392

in this normal format, the format that we are used to at this point, the f(g(x)).0399

This normally will make it easier to understand and solve problems; there are very few downsides to breaking it into this thing.0404

So, I would really recommend: any time you come up against a problem, and you are not quite sure what to do,0411

and it is this sort of thing: break it into f(g(x)), f acting on g(x)--0415

something acting on something else acting on the input that you are putting in.0420

This method, the second form of notation--this is really great as a way to look at things.0424

I really recommend that, when you see this, you break it into this thing right here; it will really help you understand what is going on.0429

Another way we can visually interpret it--this is a little hard to see what is going on here, but try to follow me0438

on what I am saying here--what we do is start with some x; we start with x, and then we apply g to that x.0444

g goes along and takes it to g(x); then, f comes along, and it hits this g(x), and it turns into f acting on g(x).0451

But we can also think of it as some new function that has been created, f composed with g.0460

We have created a new machine that can just go directly from our original x to the end result of f(g(x)).0466

It does both of these actions, both of these processes, in one thing; it is a machine that is built out of both of the machines inside of it.0476

We can look at it as stair-stepping across, or we can look at it as one new-built, giant leap,0483

where it does both of these actions in one jump.0489

All right, you can take steps across the pond, or you can take one giant leap.0492

But ultimately, they do the same thing; the leap has to be informed by the steps, though.0495

So, how do we actually use these composite functions?0502

We understand the idea behind them now; and it turns out that using them actually isn't that hard.0504

It is just important to understand the idea.0509

Each function has its own rule, like f(x) = x3 means to cube your input.0511

So, composing multiple functions just means using these rules in succession.0517

This idea is shown beautifully in that notation--that notation that I was talking about being the one I really recommend earlier, f(g(x)).0521

The function says that the function g acts on x; then, f acts on the resulting g(x).0528

So, g acts on x; that is what this says right here; and then f acts on the resulting g(x).0535

f acts on what we just had there; great.0543

Since we almost certainly know what g(x) is, and what f(x) is, from the problem, we just use that as an input for f.0546

We use the rules that we were given earlier, and we just apply them to these things.0552

Let's see an example: for example, if we had f(x) = x2 + 3 and g(x) = 2x - 2,0556

then f(g(x)) is equal to...well, what we see here--don't get tricked by the fact that we have x showing up multiple times.0564

Remember, it is just a placeholder: f(x) is just a way of saying f(whatever is in here), whatever f is acting on.0572

The thing that it is acting on will get squared; plus 3.0580

So, if it is acting on g(x), then what is g(x)? Well, g(x) is 2x - 2; so we are plugging in 2x - 2.0583

So then, we plug that in for f(x); f(x) becomes x2 + 3; so if what is inside of the box is 2x - 2,0592

it is going to be (2x - 2)2; so the box has the same process happen--it is just a new thing going on.0601

Instead of x going into it, it is just 2x - 2 going into it.0612

The same processes: it is taking the input, squaring it, and adding 3.0617

So, instead of taking an x, squaring it, and adding 3, we are taking in 2x - 2; we are squaring 2x - 2; and then we are adding 3.0621

So, if we wanted to, at this point we could expand (2x - 2)2 + 3; but this is really the key idea--0630

getting to this point of thinking of it as boxes; we are plugging in, based on boxes.0634

And we will see a bunch of examples using this idea later on.0639

But you want to think of it as we are just swapping out; we are using x as a placeholder.0642

It is not x that we are really attached to; f(x) is saying f of box, and then what happens to box;0647

f of placeholder, and then what happens to placeholder; f of input, and then what happens to input.0653

That is the way you want to think about it; and that makes it really easy to do composite functions.0659

All right, it is time for some examples.0663

So, f + g of 3; if we have f(x) = 2x + 3, and g(x) = x2 - 7, what would f + g of 3 be?0665

We do this: f(x) is 2x + 3; g(x) is x2 - 7; we have 2x + 3 + x2 - 7.0675

So, that becomes something; we could simplify it; but at this point, let's plug in x = 3.0686

We have that x = 3 is going to get plugged in, so we have 2(3) + 3 + 32 - 7.0692

6 + 3 + 9 - 7; 9 + 9 is 18; 18 - 7 is 11; so we have 11 as the answer here.0702

All right, the next one we will do with the color blue: g - f(1); what is g?0713

g is x2 - 7; what is f? it is -...and here is the key thing; it is not minus 2x; it is minus all of f;0718

not just the 2x, but minus (2x + 3); it is a whole quantity that we have to be subtracting.0728

Now, we will plug in; what happens when we plug in x = 1?0733

Well, we have 12 - 7 - (2(1) + 3); so that is 1 - 7 - 2 + 3, which is equal to -6 - 5, equals -11.0738

Great; the next one--I will do this one in green: fg(-2).0761

What is f? f is 2x + 3; and then, we are multiplying that by g; so it is that whole f, times the whole of g, x2 - 7.0766

So, (2x + 3)(x2 - 7); now let's plug in -2; when we plug in -2, we get 2(-2 + 3) (it has to be in that whole quantity),0776

times (-2)2, minus (oops, I accidentally made a plus sign) 7; great.0786

So, that is equal to 2 times -2, which is -4, plus 3; (-2)2 is 4 - 7; -4 + 3 gets us -1; 4 - 7 gets us -3; and we get positive 3.0795

Great; and finally, let's go back to red for our very last one, g/f(8).0813

So, what is g? g is x2 - 7, over all of f; so it is 2x + 3.0819

So, we plug in x = 8; 82 - 7, over 2(8) + 3; 82 is 64, minus 7, over 2 times 8 (is 16), plus 3.0827

64 - 7...we get 57, over 19; and it turns out that 57/19...19 times 2 will get up to 38; 19 times 3...we get up to 57.0844

So, 57/19...we get 3 once again, by chance; great.0854

Also, I want to point out: if we wanted to, we could have done this by figuring out what f was and what g was, separately.0859

So notice: f(3)...for example, we used the f + g(3) just to make a point here.0866

So, f(3) is equal to 2(3) + 3, which would be equal to 6 + 3, or 9.0876

g(3) is equal to 32 - 7, which equals 9 - 7, or 2; so f + g(3) is equal to, by the way we defined it, f(3) + g(3),0884

which is equal to 9, f(3), plus g(3) is 2, which equals 11; that is the exact same thing we got when we started by adding.0903

So, we can either put the functions together, and then plug in the variable;0914

or we can plug in the variable into each function and then add them together.0916

It depends on the way we want to approach it; sometimes it will be more useful to do it one way, sometimes more useful to do it the other way.0920

But it is important to notice that we can do it the other way.0924

The second example: we have the same functions that we just ended up using, f(x) = 2x + 3 and g(x) = x2 - 7.0927

What is f composed with g, first?--so we will start with the green for this one.0935

f composed with g of x; what was my recommendation? It was f(g(x)); that is the exact same thing.0939

It is another way to say this exact same thing, but it my opinion, makes it much easier to understand what it is going on.0948

f(g(x)): f(x) = 2x + 3, but we don't need that information yet; we need to plug in g(x) first.0957

f(x2 - 7); now, what is the x? Here is our x here, and x goes right here.0967

That means that f of box is equal to 2 times box plus 3.0979

The thing in the box now is x2 - 7; so that gives us f(x2 - 7) is 2 times the thing in the box, x2 - 7, plus 3.0983

And if we wanted to, we could expand that out.1000

We expand that out pretty easily; and we would get 2x2 - 14 + 3, which is 2x2 - 11; great.1003

The next one: blue for the next one: g composed with f(x) is much easier to write as g(f(x)).1015

What is f(x)? f(x) is 2x + 3, so that is what is going into g right now: 2x + 3.1024

And now, if we plug into g, g of box equals box squared minus 7.1032

So, g(2x + 3) is equal to (2x + 3)...that is the thing in the box...squared, minus 7.1040

We work that out; we get 4x2 (2x times 2x is 4x2) + 2x + 3, and (3 + 2)x is 6x, plus 6x is 12x,1048

plus 3 times 3 is 9, minus 7, which equals 4x2 + 12x + 2.1059

Great; OK, the next one--let's use red for this one: f composed with f--f composed with itself.1069

We can also write this as f(f(x)); what is f(x)? f(x) is 2x + 3, so f(2x + 3).1080

And now, the thing in our box is 2x + 3, so it is f of box equals 2 box + 3; so f(2x + 3) is 2(2x + 3) + 3.1089

Great; we just work this out: 2(2x + 3) is 4x + 6 + 3 = 4x + 9.1101

Here we are on the very last one; use green again--g composed with g(x); g(g(x))--it is much easier to see what is going on that way.1111

g(x) is x2; so now it is g acting on x2 - 7; remember, g of box is equal to box squared, minus 7.1121

So, if we are plugging in x2 - 7, it is going to be (x2 - 7)2 (remember, box squared minus 7), and then also - 7.1130

Great; so we square x2 - 7; x2 times x2 is x4;1139

x2 times -7 is 7x2, minus 7x2...another -7x2;1144

we have -7x2 + -7x2; that is minus 14x2;1149

and -7 times -7 is positive 49; and finally, minus 7; so it is x4 - 14x2 + 42; there we are.1154

There are a bunch of different function compositions, but it is not that hard, as long as we are plugging one thing into the other,1173

and remembering, in terms of the substitution: it is not about the letter x; it is about if we just had a box here.1177

If we just had a placeholder here--if we just had this thing to hold a space, then we saw what happened to that space when it was held open.1184

If we put in an input, what happens to the input?1190

We just happen to use x, because it is a convenient thing; we are used to using it as a placeholder.1192

But x isn't inherently important; it is just the idea of what happens to an input.1197

So, if we plug in something like 2x + 3, different things will happen than if we had just plugged in x.1202

Also, one other thing I want to point out: notice that, in general, f(g(x)) is not equal to (they are normally very different) g(f(x)).1207

For the most part, flipping the order that we do our function composition in gives us very, very different results.1226

Sometimes, it will end up being the same result; but mostly, if we take f(g) or g(f), it will be totally different.1234

So, mostly, f composed with g is a totally different function than g composed with f.1241

And this is just something to keep in mind--that composition order matters very much.1248

We can see it in what we got here: f(g(x)) got us 2x2 - 11, but g(f(x)) got us 4x2 + 12x + 2--totally different results.1253

So, the order that you put it in--the order that one function goes into another--the order of composition--has massive importance.1264

Great; the next example: let f(x) = x + 1, g(x) = 2x, h(x) = √(x + 1).1271

What is the domain of f, divided by g, composed with h, and then also of h composed with g composed with f?1278

Well, we will start with f(g) composed with h; now, the very first thing we want to do is figure out what f(g) is.1286

I am sorry, not f(g); what is f divided by g?1293

Well, f/g(x) is just equal to f(x)/g(x), except for when it is undefined, when g(x) is equal to 0.1296

So, what does this mean? Well, f(x) is x + 1, so (x + 1)/2x--there we are.1306

f/g(x) is equal to (x + 1)/2x; great, now we can just go back to what we are used to doing.1314

So, we use blue for this one; f/g(h(x))...it is much easier to see it written in that format.1322

f/g...what is h(x)?...f/g(√(x + 1)) = √(x + 1) + 1, over 2(√(x + 1)).1332

So, how do we figure out the domain? This right here is not our answer, but it is the function that comes out from that composition.1349

f divided by g composed with h comes out to be √(x + 1) + 1, divided by 2(√(x + 1)).1359

So, the domain is going to be everywhere where it doesn't break.1365

So, what things in here can break? Well, first, square root--any time we see square root, that breaks when a negative is inside.1373

If we have x + 1 going in in both cases, if x + 1 is less than 0 (that is to say, x + 1 is a negative value), then we have breaking.1391

It breaks--it is not defined, more formally--when x + 1 < 0, which is to say when x < -1.1400

So, that is one important point of information: x < -1 means failure.1409

Another failure point is if this bottom part is equal to 0; we have another break if 2√(x + 1) = 0.1414

And that is going to end up being x + 1 = 0, which means x = -1.1431

So, it fails if either of these conditions happens--if x is equal to -1, or x is less than -1, which is to say x ≤ -1.1436

So, its actual domain, the domain of this function, is x > -1.1450

It is all of the places where the function does not fail, where the function does not break.1459

The domain is everything that can go in; we know everything that breaks it, x < -1 and x = -1.1464

So, the domain is everything that does not break it, x > -1.1470

All right, what if we composed h with g with f?1475

All right, h composed with g composed with f might seem scary at first; but remember, we can break it into a much more pleasant, easy-to-work-with, h(g(f(x))).1479

So, first, what is f(x)? It is h of g of...what was f(x)?1490

f(x) is x + 1; so what is g(x)?...g of box is 2 times box, so g(x + 1)...it is still "h of," but g(x + 1) is going to be 2(x + 1).1495

So, let's simplify that inside just a little bit; it is h of 2...distribute; we get 2x + 2, so we have h(2x + 2), equals...1515

we plug that in here; remember, h of box is the square root of box + 1;1527

so h(2x + 2) is √(2x + 2 + 1), which is equal to √(2x + 3).1534

Great; so once again, this is not our answer; but it is going to help us figure out our answer.1549

The square root of (2x + 3) is what the function ends up being; that is what h composed with g composed with f of x is.1554

It is this thing right here; it is equal to the square root of (2x + 3).1565

So, when does √(2x + 3) break? Well, once again, it breaks--it fails--when there is a negative inside.1568

So, if 2x + 3 is negative, it breaks down; 2x + 3 < 0, so 2x < -3, which is when x is less than -3/2; we have failure.1578

It is going to be the reverse of that: everything that doesn't cause failure is the domain.1599

So, the domain is going to be everything that isn't x < -3/2, which is going to be everything greater than -3/2 or equal to it.1603

So, x is greater than or equal to -3/2, when it is big enough to not cause a negative to show up inside of that square root.1613

Great; the final example: The volume of a spherical balloon is given by volume = 4/3πr3.1625

The balloon starts being inflated at time t = 0, in seconds, and its radius, in centimeters, is given by r = 3√t.1633

OK, what does that mean? Let's try to figure it out really quickly.1641

We have a spherical balloon; well, a sphere is just a ball, so that is basically what we expect when we think of balloons.1644

This is making sense; and it is being inflated--it is being blown up; and at time t = 0 (that is just when we start), its radius is given by r = 3√t.1650

So, it starts at t = 0; and what is its radius at t = 0? r = 3√t, so 3√0, so its radius is 0.1661

So, it starts completely small; it is completely uninflated--it is just a dot at 0.1670

And then, from there, it inflates; it grows out from that point; it grows out from that moment in time.1675

Give the volume of the balloon as a function in time.1682

We blow into the balloon, and the radius expands, and the radius expands, and the radius expands.1685

And as the radius expands, there is now volume inside of the balloon.1689

What is the volume at 30 seconds?1693

All right, the first thing we need to do is give the volume of the balloon as a function of time.1696

Well, first, we might want to see these as functions, because right now, V = 4/3πr3, r = 3√t...they are not actually functions right now.1702

But we could easily turn them into functions: volume is really just a function of radius,1710

because the only thing that can vary in there is the radius.1715

Well, radius is a function based off of time, because the only thing that can vary in it is time, 3√t.1725

The volume of the balloon is a function of time; well, the volume of the balloon doesn't have time inside of it.1731

But we do know that volume has radius inside of it; and radius has time inside of it.1737

So, we can just put these together; we can compose them; and volume of radius of time will be equal to a function,1742

based off of time, that will give the volume of the balloon.1752

Let's see what that is: volume of 3√t...now we are just plugging in the radius at any given time.1755

And that is going to be the volume of 3√t; so if we plugged in our box for r, that gives us box cubed, times the other things.1763

So, it is going to be 4/3π times (3√t)3.1773

We simplify this out a bit; we get 4/3π times 33 times (√t)3.1780

Notice: 33 can cancel down to a squared and take this one out.1794

We have 4π times 32; well, 4π32...what is 32? 32 is 9.1801

4 times 9 is 36, so we have 36π.1810

What about √t3? Well, remember: √t2 (let's put it in a different color,1814

so we don't get it confused) would just be equal to t on its own; so √t3 is just one extra √t left over.1820

So, that gives us times t√t; and there we are.1830

This is the volume of this balloon, volume of radius of time; but it is also a way of seeing volume that is purely in terms of t.1834

t shows up; t shows up; but π is a constant; 36 is a constant; so what we have now is volume based purely off of time.1843

We have the first part of this question done.1852

The next part--volume at 30 seconds: well, we have two options for how to do this.1856

We could plug in, into the function that we just built, volume at 30 equals 36π times 30 times √30.1860

Or we could plug in volume of radius at time t, which would be volume of 3√30, which would be equal to 4/3π(3√30)3.1870

And it ends up being the case that these two things actually end up equaling the exact same thing.1891

Let's just fold them together: 36π times 30 times √30...that ends up being 1080π√30.1895

And if want to get this as an approximate value, something that we could actually know as a number,1907

as opposed to just having symbols that are precise and accurate, and exactly correct and right,1910

but hard to actually grasp as a single number and know what we are talking about,1915

we could get a pretty close thing, and we could round this to 8584 using a calculator.1919

What are the units that it comes in? It is centimeters cubed, because if radius is in centimeters,1926

and volume is centimeters cubed (and it makes sense, because we are talking about volume),1932

and length is centimeters, area is centimeters squared, and volume is centimeters cubed, at least if we are using centimeters.1937

If we are using meters, it is meters, meters squared, meters cubed.1944

If we are using inches or feet, it is square inches, square feet, and cubic inches and cubic feet.1947

All right, great; that completes it for composite functions.1952

I hope you have a much better understanding of what is going on.1954

Remember, when you see that circle, it means "composed with," but it is much easier1956

to break it into f of g of x, or g of f of x, depending on the order it goes in.1959

And remember, it is just going to be based off of the order that they are hitting the x in.1965

Whichever is closer goes first; so this becomes f of...g gets to act first, because it is closer to the x.1970

That is what that means; whichever is closer goes first, so it is whatever the order is with the circles.1979

But now, f(g(x)); f, circle, g(x) becomes f(g(x)); a, circle, b, circle, c(x) becomes a(b(c(x))).1984

Great; all right, I am glad to have taught you the composite functions.1995

I hope you can use it in a bunch of places; it will show up in a variety of things--it is really useful stuff here.1999

And we will see you at Educator.com later--goodbye!2003