INSTRUCTORS  Carleen Eaton Grant Fraser  Dr. Carleen Eaton

Operations on Functions

Slide Duration:

Section 1: Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
Section 2: Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
Section 3: Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
Section 4: Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
0:18
Same Dimensions
0:25
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
Section 5: Quadratic Functions and Inequalities

31m 48s

Intro
0:00
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09

27m 3s

Intro
0:00
0:16
Standard Form
0:18
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
10:12
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56

22m 48s

Intro
0:00
0:21
Standard Form
0:29
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
13:50
16:03
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23

27m 5s

Intro
0:00
0:11
Test Point
0:18
0:29
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
Section 6: Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
0:13
Like Terms and Like Monomials
0:23
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
9:23
Odd Degrees
12:51
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
8:22
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
Section 7: Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
12:23
13:29
16:07
18:18

41m 11s

Intro
0:00
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
3:24
3:47
6:33
7:16
Rationalizing Denominators
8:27
9:05
11:47
Conjugates
12:07
13:11
16:12
16:20
16:28
19:04
Distributive Property
19:10
19:20
24:11
28:43
32:00
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22

31m 27s

Intro
0:00
0:11
0:22
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
11:27
Restriction: Index is Even
11:53
12:29
15:41
17:44
20:24
24:34
Section 8: Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
Section 9: Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
16:54
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
Section 10: Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
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

47m 4s

Intro
0:00
0:22
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
12:48
13:09
21:42
29:13
35:02
40:29
Section 11: Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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• ## Related Books 0 answersPost by DJ Sai on September 15, 20186:48, confusion 1 answerLast reply by: Jerry XuFri Aug 3, 2018 11:22 PMPost by Krishna vempati on June 27, 2018im confused on division 1 answer Last reply by: Dr Carleen EatonThu Mar 27, 2014 6:40 PMPost by HEMAL SINDHVAD on March 9, 2014please this example (âˆœ27)/(âˆš3) I got answer 3^1/4 is it correct or not can you please explain if is wrong 0 answersPost by Prince Dalieh on July 10, 2013Jose Gonalez-Gigato,You are 100% right I think she makes a mistake in example 1 when solving for (f-g)(*). 0 answersPost by julius mogyorossy on June 23, 2013I was experimenting with radicals, I found that what is true in one reality, is not true in the reality, dimension of radicals, I was very surprised to learn this, what does this have to teach me about reality at large, I can't believe I could learn anything more, but maybe I am wrong. 1 answerLast reply by: Norman CervantesSun May 5, 2013 8:20 PMPost by Norman Cervantes on May 5, 201316:25, minor error, 9-6+1 should equal 4 not 2. 1 answer Last reply by: Dr Carleen EatonWed Dec 28, 2011 9:23 PMPost by Jose Gonzalez-Gigato on December 28, 2011In Example I, subtraction should be +4x^4, not -4x^4; the minus signed was incorrectly dropped when writing down the terms within parenthesis.

### Operations on Functions

• The composition of f and g exists only if the range of g is a subset of the domain of f.
• The composition of f and g is almost never equal to the composition in the reverse order.

### Operations on Functions

Let f(x) = 2x3 + 2x2 − 10x + 2 and g(x) = − 4x3 − 2x2 + 4x + 5
Find (f + g)(x),f(x) − g(x)
• Notice that operations on functions is very similar to operations on polynomials.
• When adding, combile the like temrs, when subtracting, distribute the negative sign and make it
• (f + g)(x) = f(x) + g(x)
• = ( 2x3 + 2x2 − 10x + 2 ) + ( − 4x3 − 2x2 + 4x + 5 )
• (f + g)(x) = − 2x2 − 6x + 7
• Now do subtraction, distribute the negative
• (f − g)(x) = f(x) − g(x)
• = ( 2x3 + 2x2 − 10x + 2 ) − ( − 4x3 − 2x2 + 4x + 5 )
• = ( 2x3 + 2x2 − 10x + 2 ) + ( 4x3 + 2x2 − 4x − 5 )
(f − g)(x) = 6x3 + 4x2 − 14x − 3
Let f(x) = − x3 − x2 − x − 1 and g(x) = 4x3 + 6x2 + 4x + 6
Find (f + g)(x),f(x) − g(x)
• Notice that operations on functions is very similar to operations on polynomials.
• When adding, combile the like temrs, when subtracting, distribute the negative sign and make it
• (f + g)(x) = f(x) + g(x)
• = ( − x3 − x2 − x − 1 ) + ( 4x3 + 6x2 + 4x + 6 )
• (f + g)(x) = 3x2 + 5x2 + 3x + 5
• Now do subtraction, distribute the negative
• (f − g)(x) = f(x) − g(x)
• = ( − x3 − x2 − x − 1 ) − ( 4x3 + 6x2 + 4x + 6 )
• = ( − x3 − x2 − x − 1 ) + ( − 4x3 − 6x2 − 4x − 6 )
(f − g)(x) = − 5x3 − 7x2 − 5x − 7
Let g(x) = 2x2 − 2x;h(x) = − 2x3 + 2x
Find (g*h)(x) and ([f/g])(x)
• To multiply, use the foil method, or the distributive method or the box method as covered in previous
• excercises.
• (g*h)(x) = g(x)*h(x) = (2x2 − 2x)( − 2x3 + 2x) = 2x2( − 2x3) + 2x2(2x) − 2x( − 2x3) − 2x(2x)
• Simplify
• (g*h)(x) = − 4x5 + 4x3 + 4x4 − 4x2 = − 4x5 + 4x4 + 4x3 − 4x2
• Divide, factor if necessary. State the constraints in the domain.
• ([f/g])(x) = [f(x)/g(x)] = [(2x2 − 2x)/( − 2x3 + 2x)] = [(2x(x − 2))/(2x( − x2 + 1))] = [((x − 2))/(( − x2 + 1))] = [(x − 2)/( − x2 + 1)]
• To find the restrictions, set the denominator equal to zero.
• − x2 + 1 = 0
• x2 = 1
• x = ±1
([f/g])(x) = [(x − 2)/( − x2 + 1)] except when x = 1,x = − 1
Let g(x) = 2x − 4;h(x) = 4x + 2
Find (g °h)(x),(h °g)(x)
• This is called Function Composition.
• Every where you see an x in g(x) must be replaced by h(x)
• (g °h)(x) = g(h(x)) = 2(h(x) − 4 = 2(4x + 2) − 4
• Simplify
• g(h(x)) = 8x + 4 − 4 = 8x
• To find (h °g)(x),h(g(x)),
• every where you see an x in h(x) must be replaced by g(x)
• (h °g)(x) = h(g(x)) = 4(g(x)) + 2 = 4(2x − 4) + 2
• Simplify
(h °g)(x) = h(g(x)) = 8x − 16 + 2 = 8x − 14
Let g(x) = 2x − 3;h(x) = x3 + 2x2
Find (g °h)(x),(h °g)(x)
• This is called Function Composition.
• Every where you see an x in g(x) must be replaced by h(x)
• (g °h)(x) = g(h(x)) = 2(h(x)) − 3 = 2(x3 + 2x2) − 3
• Simplify
• g(h(x)) = 2x3 + 4x2 − 3
• To find (h °g)(x),h(g(x)),
• every where you see an x in h(x) must be replaced by g(x)
• (h °g)(x) = h(g(x)) = ((g(x))3 + 2(g(x))2 = (2x − 3)3 + 2(2x − 3)
• Simplify
• (h °g)(x) = (2x − 3)(2x − 3)(2x − 3) + 4x2 − 6 = 8x3 − 36x2 + 54x − 27 + 4x2 − 6 =
(h °g)(x) = 8x3 − 32x2 + 54x − 33
Let g(x) = x2 − 2x;h(x) = 3x + 1
Find (g °h)(x),(h °g)(x)
• This is called Function Composition.
• Every where you see an x in g(x) must be replaced by h(x)
• (g °h)(x) = g(h(x)) = (h(x))2 − 2(h(x)) = (3x + 1)2 − 2(3x + 1)
• Simplify
• g(h(x)) = (3x + 1)2 − 2(3x + 1) = 9x2 + 6x + 1 − 6x − 2 = 9x2 − 1
• To find (h °g)(x),h(g(x)),
• every where you see an x in h(x) must be replaced by g(x)
• (h °g)(x) = h(g(x)) = 3(g(x)) + 1 = 3(x2 − 2x) + 1
• Simplify
(h °g)(x) = h(g(x)) = 3(x2 − 2x) + 1 = 3x2 − 6x + 1 =
Let g(x) = x − 1;h(x) = x2 − 1
Find (g °h)(x),(h °g)(x)
• This is called Function Composition.
• Every where you see an x in g(x) must be replaced by h(x)
• (g °h)(x) = g(h(x)) = (h(x)) − 1 = (x2 − 1) − 1 =
• Simplify
• g(h(x)) = x2 − 2
• To find (h °g)(x),h(g(x)),
• every where you see an x in h(x) must be replaced by g(x)
• (h °g)(x) = h(g(x)) = ((g(x))2 − 1 = (x − 1)2 − 1
• Simplify
(h °g)(x) = (x − 1)2 − 1 = x2 − 2x + 1 − 1 = x2 − 2x
Let f(x) = − 4x + 2;g(x) = x2 + 5x
Find (fog)(3)
• You are now evaluating a function composition at a value.
• Evaluate g(3), and the result will become the input to the function f.
• Evaluate f at g(3)
• (fog)(3) = f(g(3)) = f((3)2 + 5(3)) = f(9 + 15) = f(24) = − 4(24) + 2 = − 96 + 2 = − 94
f(g(3)) = − 94
Let f(x) = − 3x − 1;g(x) = x3 − 5x2
Find (fog)(0)
• You are now evaluating a function composition at a value.
• Evaluate g(3), and the result will become the input to the function f.
• Evaluate f at g(3)
• (fog)(3) = f(g(0)) = f((0)3 − 5(0)2) = f(0) = − 3(0) − 1 = − 1
f(g(0)) = − 1
Let f(x) = − 3x − 1;g(x) = x3 − 5x2;h(x) = 2x − 1
Find f(g(h(1)))
• You are now evaluating a function composition at a value.
• Evaluate h(1), and the result will become the input to the function g(x).
• Evaluate g at h(1) and the output will become the input for the function f(x).
• f(g(h(1))) = f(g(2(1) − 1)) = f(g(1)) = f((1)3 − 5(1)2) = f(1 − 5) = f( − 4) = − 3( − 4) − 1 = 12 − 1 = 11
f(g(h(1))) = 11

*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.

### Operations on 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
• Arithmetic Operations 0:07
• Domain
• Intersection
• Denominator is Zero
• Example: Operations
• Composition of Functions 7:18
• Notation
• Right to Left
• Example: Composition
• Composition is Not Commutative 17:23
• Example: Not Commutative
• Example 1: Function Operations 20:55
• Example 2: Function Operations 24:34
• Example 3: Compositions 27:51
• Example 4: Function Operations 31:09

### Transcription: Operations on Functions

Welcome to Educator.com.0000

Today, we are going to be discussing operations on functions, beginning with arithmetic operations.0002

Recall that two functions can be added, subtracted, multiplied, or divided.0010

The domain of the sum, difference, product, or quotient is the intersections of the domains of the two functions.0016

And remember that intersection, when you are talking about sets, is the areas of the sets that overlap.0024

Often, we will be talking about domains that are all real numbers, and so then there would be complete overlap of those two.0031

But if the sets are slightly different (the domains are slightly different), the domain of these is only0038

the intersection of the domains of the two functions you are performing the operation on.0044

We will talk about when we are working with division, because that is a special case, and there are some additional restrictions on the domain.0049

Let's take, as an example, the two functions f(x) = 2x2 + 5, and a second function g(x) = 3x2 - 4.0062

OK, if you were to add these, then the notation would be as follows: f + g(x).0085

And this can be rewritten to show that it really means f(x) + g(x).0096

So, f + g(x) is just the two functions added together.0104

Looking at what my two functions are: I have 2x2 + 5 as the first function,0110

and I am going to be adding that to my second function, which is 3x2 - 4.0122

And recall that, when you are adding polynomials, all you need to do for addition is remove the parentheses.0128

And that, when we go ahead and add like terms, is going to give us 5x2.0139

And then, combining the constants, 5 - 4 is 1.0144

So, the sum of these two functions is 5x2 + 1.0148

OK, now talking about subtraction: if we were to subtract f - g(x), this can be considered as f(x) - g(x).0155

So, it is the same idea as addition, but you just have to be more careful with the signs.0169

So here, we are going to be taking (2x2 + 5) - (3x2 - 4).0176

Now, when you remove the parentheses, for the first one--this first expression here--this is really positive in front of it;0186

so removing the parentheses is simple--you just take them away; the signs stay the same.0197

To remove the parentheses for this second expression, you have to reverse the signs.0201

So, when I apply the negative sign to 3x2, that is going to give -3x2.0206

Applying the negative sign to the -4 will give me positive 4.0212

And then, I combine these: I get like terms, 2x2 - 3x2, to get -x2; and 4 + 5 (5 + 4) gives me 9.0217

So, that was addition and subtraction of two functions; now, let's talk about multiplication.0231

f times g(x) is the same as f(x) times g(x); this is the same as multiplying two binomials, as we have done in earlier work in this course.0241

So, it is (2x2 + 5) times (3x2 - 4); and since these are binomials,0262

I use the FOIL method: multiplying the first two terms gives me 4 times 3 is 6x4;0271

my outer terms are going to give me -8x2; the inner two terms...5 times 3...that is 15x2.0280

And finally, 5 times -4 gets -20.0291

Simplify by combining like terms: I combine the two x2 terms, and that is going to give me...0295

-8x2 + 15x2, is going to give me positive 7x2; minus 20.0302

OK, so this is multiplication of two functions.0311

Finally, division: division has that additional restriction on the domain, as I mentioned.0317

Let's look at dividing here: f divided by g(x), which is actually the same as f(x) over g(x).0324

f(x) is 2x2 + 5...over 3x2 - 4.0338

Now, when you are handling division, you need to think about excluded values in the denominator.0347

As always, if the denominator is 0, that would result in an expression that is undefined.0354

You cannot divide by 0, so we have to think about situations where this expression,0360

3x2 - 4, would equal 0, and then exclude those from the domain.0366

So, I am going to go ahead and solve for x, and see what values of x would make this expression 0.0371

I can do that by adding 4 to both sides, dividing both sides by 3, and then taking the square root of both sides.0378

This tells me that the excluded values, when performing this operation (performing division and finding the quotient) are x = √ or -√4/3.0394

So, x cannot equal ±√4/3, because if it does, the result will be a 0 in the denominator and a value that is undefined.0419

So, this is a situation that applies only to division.0430

Composition of functions: suppose that f and g are functions, and the range of g, the second function, is a subset of the domain of f, the first function.0438

I will talk a little bit more about that in a second.0453

But right now, let's look at what the notation is, and what it is really saying when we talk about composition of functions.0456

The composition of f and g is defined by...this symbol, this open circle, means "composed with."0465

It is not a multiplication symbol: the open circle would be read as "f composed with g of x."0476

And that is going to result in a composite function.0483

And another way to write this, and to think about it, is that what this is saying is f(g(x)).0486

Something to keep in mind as we work with composition of functions is that you actually move from right to left when you are working with these.0493

So, when I consider a composite function (and it could actually be composed of three functions or four functions),0502

I am going to start out by working my way from the right, seeing if I am given a value here--0509

instead of g(x), it could be g(2)--and then working with that, determining what g(x), and then plugging that value into my f function.0515

OK, so let's first use an example: if I have two functions--one is f(x) = x2 - 2x + 1,0528

and another one is g(x) = 4x - 5--I might be asked to find the composition of those two functions.0538

And that would be...I could be asked to find f composed with g, or g composed with f; but I am going to work with f composed with g.0548

Again, I am rewriting that as...this is saying it is f of g of x.0564

So, I am going to start here with this function on the right.0571

And I am going to say, "OK, this is really saying f(g(x))."0574

Well, I look here, and g(x) is 4x - 5.0577

Previously, we have talked about evaluating a function for an algebraic expression.0585

We first started out by evaluating functions for numbers, like f(3): I would just substitute all of the x's with 3's.0591

I also talked about finding f for an algebraic expression; and that is really what we end up doing here.0601

So, if I am trying to find f(4x - 5), wherever I see an x, I just replace it with this expression, 4x - 5...0606

which is going to be...here I have x2, so I am going to replace that x with 4x - 5 and square it.0615

I have a second x right here; I have -2 times whatever is in here, since that is replacing x.0625

And then, I am adding 1; so right here, where there is an x, simply substitute 4x - 5.0633

And then, we are going to figure what this would become equal to.0641

And this is going to give me...if you recall, if I square a binomial, if I form a perfect square trinomial from that...0645

it ends up being this first term squared (4x, squared), and then 2 times the product of these two terms;0658

so this is going to give me 4x, times -5, plus the square of this last term.0669

You could always write this out as (4x - 5) (4x - 5) and use FOIL if you didn't remember that rule.0678

OK, then over here, I am going to multiply everything inside the parentheses by -2 to get -8x + 10 + 1.0685

OK, so this is going to give me...I am actually rewriting this like that, because I also have to square the 4.0698

So, this is going to give me (4x)2, so that will actually be 16x2,0709

and then 2 times 4x is 8x, times -5 is going to give me 8x times -5...-40x.0717

And then here, I have -5 times -5, so that is + 25, minus 8x plus 10 plus 1.0734

Simplifying, I still have 15x2; combining -40x and -8x is going to be -48x.0743

25 + 10 + 1 is 35...36.0753

f composed with g: I went about finding this by first replacing this with g(x), which is 4x - 5,0762

and then finding f of that by replacing all of the x's here with 4x - 5.0769

And then, I squared this binomial, took 2 times the binomial here, and then added 1.0776

And this is the result: f composed with g of x equals this.0783

Now, talking a little bit more about domain and range: let's say that we were asked to find f composed with g(2).0790

Well, I know what f composed with g(x) is, and I could just plug the 2 in here.0802

But let's say we hadn't done that work--that we were just given these two, and then told, "OK, find f composed with g(2)."0808

So, I am just going to start there and think about how this works.0814

Well, I start at the right; I am going to rewrite this here, first, as f(g(x)).0818

In this case, I said that x is going to be 2; so I am going to put a 2 there.0828

So, I am going to find g(2), and then I am going to find f of that value.0833

So, this is going to equal f of...well, g(x) is 4x - 5, but I am asked to find g(2).0838

So remember that g(2) is going to be 4 times 2, minus 5.0847

I am going to put that in here: 4 times 2, minus 5.0854

OK, that equals f of 8 - 5, which is 3, so I am being asked to find f(3).0863

Now, let's think about this for a second: here, my result, 3, is an element of the range of g.0874

My input value for g--my domain value--is 2; I evaluated that for this function g, and I came up with 3.0894

g(2) is 3, so 3 is part of the range.0903

Now, I am using that as my input value for the function f.0908

So, 3 is an element of the range of g, and an element of the domain of f.0914

The range of g is a subset of the domain of f; that is what this is saying up here.0928

So, I evaluate g for a particular value--just the general case x, or a certain number.0934

I find what that is: that is a part of this range; it is also an element of the domain of f.0942

The entire set of values--the entire range for g--is a subset of the domain of f.0950

So, let's go ahead and finish this out: I found g(2) to be 3; now I am going to evaluate f for 3.0963

Well, f is x2 - 2x + 1, so wherever I see an x, I am simply going to substitute in a 3.0972

That is going to give me 9 - (6 + 1); that is 9 - 7, so that is 2.0982

So, f composed with g(2)...actually, that is 4--a correction right there: f composed with g(2) is 4.0992

So again, starting out, we are going right to left, finding g(2), substituting what I came up with,1013

which is 3, right in here, finding f(3), and evaluating for that.1024

And I find that f composed with g(2) is 4.1029

Up here, I just talked about the general case, f composed with g(x), and I found this.1034

OK, so that is an introduction to composition of functions.1040

And one important thing to keep in mind is that composition is not commutative.1043

What we mean by that is that, in general, f composed with g does not equal g composed with f.1048

but in general, you can't just flip these two around and assume that the opposite case is equal, because it is actually usually not.1061

Illustrating that with an example: let's let f(x) equals x2 + 1, and g(x) equal 2x - 3.1071

Now, I am going to find both f composed with g and g composed with f, and then compare what I get.1091

So, f composed with g(x) = f(g(x)); starting from the right, let's put g(x) in here, which is 2x - 3.1096

Now, I am evaluating f for this expression, 2x - 3.1115

And f is x2 + 1, so I am going to substitute 2x - 3 for this x, and square it, and then add 1.1120

Squaring this would give me 2x squared, plus 2 times 2x times -3, plus -3 squared, plus 1.1133

OK, that gives me 4x2; 2 times 2 is 4, times -3 is -12x; plus -32, which is 9, plus 1.1150

Simplify to 4x2 - 12x + 10.1166

f composed with g(x) is equal to this; now, let's try g composed with f(x), which is equal to g(f(x)).1173

f(x) is x2 + 1; I am going to put that in here.1190

Now, I am going to evaluate g for x2 + 1 by substituting x2 + 1 everywhere there is an x in this function.1195

That is going to give me 2 times x2 + 1 - 3.1206

2 times x2 is 2x2; plus 2 times 1 is 2; minus 3 gives me 2x2 - 1.1217

And this is g composed with f(x).1227

As you can see, f composed with g(x) is not the same as g composed with f.1232

So, I can't make that assumption, that these two are the same.1245

In fact, if anything, it is likely that they are not the same; composition is not commutative.1248

The first example: in this example, we are given two functions, f(x) and g(x), and asked to find their sum and their difference.1257

Starting out with addition: recall that f + g(x) = f(x) + g(x).1265

So, I am simply going to add the two functions; f(x) is 3x3 - 2x2 - x - 1,1274

plus -4x4 + 2x3 - 3x - 4.1286

OK, since this is addition, I can simply remove the parentheses without worrying about having a problem with the signs.1294

The signs remain the same, so I just continue to retain the signs in here, since this is addition.1302

I can just take the parentheses away.1313

Let's add this, and at the same time put it in descending order to help keep track of everything.1320

The largest power I have is 4; so I have -4x4, and there is no like term that I can combine that with.1326

For terms that are cubed, I have 3x3 and 2x3 to give me 5x3.1335

For terms that are squared, I only have one term, and that is -2x2.1342

For x's, I have -x and -3x to give -4x; and for constants, -1 and -4 are -5.1348

So, f + g(x) equals this expression.1360

Subtraction: f - g(x) = f(x) - g(x); now here, I do need to be careful with the signs.1370

So, I am rewriting: 3x3 - 2x2 - x - 1, minus g(x), which is 4x4 + 2x3 - 3x - 4.1384

OK, removing the parentheses: since this is a positive in front of this, it remains 3x3...1398

Here, I need to apply the negative sign to each term inside the parentheses; and I can do that by reversing the sign.1413

So, this becomes -4x4; -2x3; a negative and a negative--that gives me a positive 3x; and a negative and a negative...plus 4.1419

Combining like terms, I get -4x4; here, I have 3x3 and -2x3, leaving + x3.1432

I only have one x2 term, so I leave that alone; that is -2x2.1447

For x's, -x + 3x is going to leave me with just 2x, and then the constants: -1 + 4 gives me _ 3.1453

So, f - g(x) is given by this expression right here.1466

Example 2: this time, we are given two functions, f(x) and g(x), and told to find the product and the quotient of these functions.1474

OK, beginning with multiplication: the product f times g(x) equals f(x) times g(x),1485

which equals (4x2 - 7) times (3x2 + 9x).1499

So, the first terms--multiplying those, that is 12x4; the outer terms give me 4 times 9; that is 36x2 times x, so that is 36x3.1513

The inner two terms--that is -21x2; and then -7 times 9 is going to give me -63; and I have an x here.1528

OK, and looking at this, I can't simplify this any further, because I don't have any like terms.1539

So, I am just going to leave that alone; and this gives me f times g of x.1547

OK, the second task is to find the quotient, f divided by g of x, which equals f(x) divided by g(x).1553

which equals (4x2 - 7) divided by (3x2 + 9x).1567

Now, recall: when working with division of functions, we need to find the excluded values.1578

And the excluded values are those values of x, those numbers of the domain, that will make the denominator 0,1582

because if this denominator is 0, I will have an undefined expression.1590

So, any value of x that makes 3x2 + 9x equal 0 is excluded from the domain.1594

Handling this by factoring: I can see that I have a greatest common factor of 3x.1604

I have a 3 here and a 3 in here, and I have an x here and an x in here that I can pull out.1611

This will leave behind an x and a 3.1615

Using the zero product property, this tells me that 0 = 3x, or it could be that x + 3 = 0.1620

If either of those is true, this product becomes 0.1633

OK, so this will end up giving me (let's rewrite this as) 3x = 0, so divide both sides by 3: x = 0--that is an excluded value.1638

And x = -3--these are excluded values, meaning this function is not defined for domain values of x = 0 and x = -3.1652

Those values are excluded from the domain.1667

Example 3: now, we are going to work with composition of functions.1671

Given f(x) and g(x), we are asked to find f composed with g(x) and g composed with f(x), starting with this one.1677

OK, so recall that f composed with g(x) is the same as saying f(g(x)).1686

So, I am going to start from the right and go to the left.1698

First, I am going to figure out what g(x) is; and I have that given right here as 4x2.1702

I am asked to find f of that; I am going to evaluate f for 4x2.1710

So, wherever I see an x in f(x), in this function, I will substitute 4x2.1715

So, this is going to give me 2 times x; and in this case, x is 4x2; minus 3.1721

What I have here is rewritten here, just with 4x2 replacing the x.1729

OK, that is going to give me 8x2 - 3; so, f composed with g of x equals 8x2 - 3.1737

Now, g composed with f(x) equals g(f(x)), starting from the right: what do I have in here?1753

I have f(x), which is 2x - 3; so this equals g(2x - 3).1766

This means I need to evaluate the function g when x is defined as 2x - 3.1774

So, I have an x here; it is 4 times x2, so 4(2x - 3)2, just replacing this with this.1785

OK, squaring 2x - 3 and multiplying that times 4 is going to give me (2x)2 + 2 times 2x times -3 + (-3)2.1795

That is going to give me 4 times...this is 2 times 2...that is going to give me 4x2;1816

2 times 2x is 4x, times -3; that is -12x; (-3)2 is 9.1826

Multiplying each of these by 4 gives me 16x2; 4 times -12 is -48x; 4 times 9 is 36.1833

And this also illustrates what we talked about before, that composition is not commutative.1846

f composed with g of x is not equal to g composed with f of x.1854

It may be, but it is not necessarily true; you cannot assume that it is true.1864

This time, we are going to be working with three functions: f(x), g(x), and h(x).1870

And we handle these the same way as we did composition of functions, when we were only working with two functions.1876

And we are going to work from right to left.1884

Here, instead of just asking for f or g or h of x, they are asking me for f(g(h(the particular value -2))).1887

OK, so I am going to start out by looking for h(-2).1897

Let's go ahead and find that value; I want to find f(g(h(-2))).1906

So, what is h(-2)? Well, h is 2x2 + 3; therefore, h(-2)...I would have to substitute in 2...I put a -2 here and square it...+ 3.1913

So, this is f(g(2 times -2 squared, plus 3)), which equals f(g(...-2 squared is 4, plus 3)).1926

So, this gives me 8 + 3, which is f(g(11)); so starting from the right, I evaluated h for -2; that gave me 11.1952

Now, this is a member of the range of h, and it is also an element of the domain of g, because I am plugging it in here as an input value.1971

So now, I need to find g(11): well, g(x) equals 4x - 2; so substituting 4 times 11 - 2 is what I am going to be doing here.1981

f(4 times 11 minus 2) equals 44 - 2, equals 42; so, g(11) = 42.2002

Finally, I am just left with f(42); so I need to evaluate f(x) when x is 42.2020

So, I simply substitute in, and that is going to give me 6 times 42 (which is 252).2031

So, the result from this composition of functions is 252, from evaluating h for -2; finding that result;2043

evaluating g for that result (which was h(-2), which was 11); evaluating g for 11;2052

finding that that was 42; and then evaluating f for that value.2060

That concludes this session of Educator.com; thanks for visiting!2066

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