INSTRUCTORS  Carleen Eaton Grant Fraser  Dr. Carleen Eaton

Relations and 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 Munqiz Minhas on August 25, 2016Do you have any videos on Interval Notations and Sets? 1 answer Last reply by: Dr Carleen EatonMon Apr 9, 2012 6:32 PMPost by jeremiah dulla on April 7, 2012Do you have any videos on graphing Functions and like Transformations of Functions Horizontal Compression,expansion etc. 0 answersPost by Jasmine Valdovinos on August 18, 2011why did you place a 3 on the domain shouldnt it be -1,0,1,2... 1 answer Last reply by: Dr Carleen EatonSat May 28, 2011 11:09 PMPost by Victoria Jobst on May 28, 2011Is example 2 discrete or continuous?

### Relations and Functions

• Each relation or function has a domain and a range.
• Functions may be one to one, and may be discrete or continuous.
• Functions satisfy the vertical line test: any vertical line crosses the graph at most once.
• One variable, usually x, is the independent variable. The other variable is the dependent variable.

### Relations and Functions

Given the relation R = { (5, − 2),(1,2),(3,2),(1,1),(2, − 1)} . Give the Domain and Range. Is R a function?
• Domain = all x's taken up by the relation
• Range = all the y's taken up by the relation
• Domain = { 1,2,3,5}
• Range = { − 2, − 1,1,2}
• Create a Map to check if the relation is a Function.
•  Domain 1 2 3 5 Maps To 1 & 2 -2 2 -2
The relation R is not a function. There are two arrows coming out from the same x = 1
Given the relation R = { (2, − 3),(4,0),(3,2),(5,1),(2,2)} . Give the Domain and Range. Is R a function?
• Domain = all x's taken up by the relation
• Range = all the y's taken up by the relation
• Domain = { 2,3,4,5}
• Range = { − 3,0,1,2}
• Create a Map to check if the relation is a Function.
•  Domain 2 3 4 5 Maps To -3 & 2 2 0 1
The relation R is not a function. There are two arrows coming out from the same x = 2
Given the relation R = { (1, − 1),(3,1),(4,2),(5,0),(6,2)} . Give the Domain and Range. Is R a function?
• Domain = all x's taken up by the relation
• Range = all the y's taken up by the relation
• Domain = { 1,3,4,5,6}
• Range = { − 1,0,1,2}
• Create a Map to check if the relation is a Function.
•  Domain 1 3 4 5 6 Maps To -2 1 2 0 2
The relation R is a function. There are no two arrows coming out from the same value of x.
The relation R is given by the equation y = 2x + 1. Is R a function? What is the domain and range?
Is R discrete or continuous?
• Create a table of values to check if the relation is a function.
•  x y=2x+1 -2 y=2(-2)+1=-3 -1 y=2(-1)+1=-1 0 y=2(0)+1=1 1 y=2(1)+1=3
• The relation is a function because it is 1 - to − 1
• Find the Domain
• Find the Range
• Domain = All real values
• Range = All real values
The relation R is continous becuase it represents a linear equation.
Graph the relation R given by 4x − 2y = 8. Is R a function? What is the domain and range?
Is R discrete or continuous?
•  x 4x-2y=8 -1 4(-1)-2y=8 -2y=12 y=-6 0 4(0)-2y=8 -2y=8 y=-4 2 4(2)-2y=8 8-2y=8 -2y=0 y=0 3 4(3)-2y=8 12-2y=8 -2y=-4 y=2
• Find the Domain
• Find the Range
• Domain = All real values
• Range = All real values
The relation R is continous becuase it represents a linear equation.
Let f(x) = 3x2 + 5x find f( − 1),f(2k)
• To evaluate the given function at the given values, substitute the input where ever there is an x.
• f( − 1) = 3( − 1)2 + 5( − 1)
• f( − 1) = 3(1) − 5
• f( − 1) = − 2
• f(2k) = 3(2k)2 + 5(2k)
• f(2k) = 3(4k2) + 10k
• f(2k) = 12k2 + 10k
f( − 1) = − 2
f(2k) = 12k2 + 10k
Let f(x) = x3 + x2 find f( − 1),f(3k)
• To evaluate the given function at the given values, substitute the input where ever there is an x.
• f( − 1) = ( − 1)3 + ( − 1)2
• f( − 1) = ( − 1)( − 1)( − 1) + ( − 1)( − 1)
• f( − 1) = − 1 + 1 = 0
• f(3k) = (3k)3 + (3k)2
• f(3k) = (3k)(3k)(3k) + (3k)(3k)
• f(3k) = 27k3 + 9k2
f( − 1) = 0
f(3k) = 27k3 + 9k2
Let f(x) = − x3 − x2 find f( − 1),f(2)
• To evaluate the given function at the given values, substitute the input where ever there is an x.
• f( − 1) = − ( − 1)3 − ( − 1)2
• f( − 1) = − ( − 1)( − 1)( − 1) − ( − 1)( − 1)
• f( − 1) = − ( − 1) − (1) = 0
• f(2) = − (2)3 − (2)2
• f(2) = − (2)(2)(2) − (2)(2)
• f(2) = − (8) − (4) = − 12
f( − 1) = 0
f(2) = − 12
Let f(x) = x4 + x2 find f( − 1),f( − k)
• To evaluate the given function at the given values, substitute the input where ever there is an x.
• f( − 1) = ( − 1)4 + ( − 1)2
• f( − 1) = ( − 1)( − 1)( − 1)( − 1) + ( − 1)( − 1)
• f( − 1) = (1) + (1) = 2
• f( − k) = ( − k)4 + ( − k)2
• f( − k) = ( − k)( − k)( − k)( − k) + ( − k)( − k)
• f( − k) = k4 + k2
f( − 1) = 2
f( − k) = k4 + k2
Let f(x) = x5 + x3 find f( − 1),f( − k)
• To evaluate the given function at the given values, substitute the input where ever there is an x.
• f( − 1) = ( − 1)5 + ( − 1)3
• f( − 1) = ( − 1)( − 1)( − 1)( − 1)( − 1) + ( − 1)( − 1)( − 1)
• f( − 1) = ( − 1) + ( − 1) = − 2
• f( − k) = ( − k)5 + ( − k)3
• f( − k) = ( − k)( − k)( − k)( − k)( − k) + ( − k)( − k)( − k)
• f( − k) = − k5 − k3
f( − 1) = − 2
f( − k) = − k5 − k3
f(x) = x5 + x4 + x3 find f( − 1),f( − 2k)
• To evaluate the given function at the given values, substitute the input where ever there is an x.
• f( − 1) = ( − 1)5 + ( − 1)4 + ( − 1)3
• f( − 1) = ( − 1)( − 1)( − 1)( − 1)( − 1) + ( − 1)( − 1)( − 1)( − 1) + ( − 1)( − 1)( − 1)
• f( − 1) = ( − 1) + (1) + ( − 1) = − 1
• f( − k) = ( − k)5 + ( − k)4 + ( − k)3
• f( − k) = ( − k)( − k)( − k)( − k)( − k) + ( − k)( − k)( − k)( − k) + ( − k)( − k)( − k)
• f( − k) = − k5 + k4 − k3
f( − 1) = − 1
f( − k) = − k5 + k4 − k3

*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.

### Relations and 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
• Coordinate Plane 0:20
• X-Coordinate and Y-Coordinate
• Example: Coordinate Pairs
• Relations 2:14
• Domain and Range
• Set of Ordered Pairs
• As a Table
• Functions 4:21
• One Element in Range
• Example: Mapping
• Example: Table and Map
• One-to-One Functions 8:01
• Example: One-to-One
• Example: Not One-to-One
• Graphs of Relations 11:01
• Discrete and Continuous
• Example: Discrete
• Example: Continous
• Vertical Line Test 14:09
• Example: S Curve
• Example: Function
• Equations, Relations, and Functions 17:03
• Independent Variable and Dependent Variable
• Function Notation 19:11
• Example: Function Notation
• 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

### Transcription: Relations and Functions

Welcome to Educator.com.0000

For today's Algebra II lesson, we are going to be discussing relations and functions.0002

And recall that some of these concepts were discussed in Algebra I, so this is a review.0008

And if you need a more detailed review, check out the Algebra I lectures here at Educator.0013

Beginning with the concept of the coordinate plane: the coordinate plane describes each point as an ordered pair of numbers (x,y).0021

The first number is the x-coordinate, and the second is the y-coordinate.0031

For example, consider the ordered pair (-4,-2): this is describing a point on the coordinate plane with an x-coordinate of -4 and a y-coordinate of -2.0036

Or the pair (0,2): the x-value would be 0, and the y-value would be 2; this is the point (0,2) on the coordinate plane.0057

Or (3,5): x is 3; y is 5.0073

Also, recall that the quadrants are labeled with the Roman numerals: I, and then (going counterclockwise) quadrant II, quadrant III, and quadrant IV.0080

In the coordinate pairs in the first quadrant, the x is positive, as is the y.0097

In the second quadrant, you will have a negative value for x and a positive value for y, such as (-2,4)--that would be an example.0103

In the third quadrant, both x and y are negative; and then, in the fourth quadrant, x is positive; y is negative.0114

And we will be using the coordinate plane frequently in these lessons, in order to graph various equations.0127

Recall that a relation is a set of ordered pairs.0135

The domain of the relation is the set of all the first coordinates, and the range is the set of all the second coordinates.0140

A relation is often written as a set of ordered pairs, using braces to denote that this is a set,0149

and then the ordered pairs, each in parentheses, separated by a comma.0158

Sometimes, the relation is represented as a table; so, -2, 1; -1, 0; 0, 1; and 1, 2.0172

We will be doing some graphing of relations also, in just a little bit.0192

So, as discussed up here, the domain is the set of all first coordinates.0196

And the set of all first coordinates here would be {-2, -1, 0, 1}.0207

The range is the set of the second coordinates; so the range right here--all the y-values--is {1, 0, 1, 2}.0219

However, you don't actually need to write the 1 twice; so in actuality, it would be written as such.0232

It is OK to repeat the values if you want, but usually, we just write each value in the domain or range once; each is represented once.0245

Functions are a certain type of relation: so, all functions are relations, but not all relations are functions.0261

A function is a relation in which each element of the domain is paired with exactly one element of the range.0272

For example, consider the relation shown: {(1,4), (2,5), (3,8), (4,10)}.0280

Each member of the domain corresponds to exactly one element of the range.0298

We don't have a situation where it is saying {(1,4), (1,6), (1,8)}, where that member of the domain is paired with multiple members of the range.0305

Another way, again, to represent this as a table--another method that can be used--is mapping.0314

And mapping is a visual device that can help you to determine if you have a function or not,0321

by showing how each element of the domain is paired with an element of the range.0327

A map would look something like this: over here, I am going to put the elements of the domain, 1, 2, 3, and 8;0333

over here, the elements of the range: 4, 5, 8, and 10.0347

And then, using arrows, I am going to show the relationship between the two.0355

So, 1 corresponds to 4 (or is paired with 4); 2 to 5; 3 to 8; and (this should actually be 4) 4 to 10.0360

OK, so as you can see, there is only one arrow going from each element of the domain to each element of the range.0374

And that tells me that I do have a function.0383

Let's look at a different situation, using a table form: let's look at a second relation.0386

In this one, I am going to have (-2,2), (-3,2), (-4,5), and (-6,7).0392

And I am going to go ahead and map this: -2, -3, -4, and -6: these are my elements of the domain.0403

For the range, I don't have to write 2 twice; I am just going to write it once; 5, and 7.0416

OK, -2 corresponds to 2; -3 also corresponds to 2; -4 corresponds to 5; and -6 corresponds to 7.0422

This is also a function, so both of these are relations, and they are also functions.0436

It is OK for two elements of the domain to be paired with the same element of the range; this is allowed.0445

What is not allowed is if I were to have a situation where I had {1, 2, 3}, {4, 5, 6}; and I had 1 paired with 4, and 1 paired with 5.0452

So, if you have two arrows coming off an element of the domain, then this is not a function.0468

Here are two examples of relations that are also functions.0476

There is a specific type of function that is called a one-to-one function.0482

And a function is one-to-one if distinct elements of the domain are paired with distinct elements of the range.0486

In the previous example, we saw a situation where we did have a one-to-one function, and another situation where we did not.0494

OK, so to review: the ordered pairs in that first function that we just discussed were (1,4), (2,5), (3,8), and (4,10).0501

OK, and we can use mapping, again, to determine what the situation is with this relation (which is also a function).0519

The domain is {1, 2, 3, 4}; and the range is {4, 5, 8, 10}.0529

When I put my arrows to show this relationship, you see that distinct elements of the domain are paired with distinct elements in the range.0540

1 is paired with 4; they are each unique--each pair is unique.0552

Looking at the other function that we discussed: the pairs are (-2,3), (-3,2), (-4,3)...slightly different, but the same general concept...slightly different, though.0558

OK, here I have -2, -3, -4, and -6; over here, in the range, I have 3, 2...I am not going to repeat the 3--I already have that...and then 7.0587

-2 corresponds to 3; -3 corresponds to 2; -4 also corresponds to 3; -6 corresponds to 7.0605

This is still a function; OK, so these are both functions: function, function.0620

However, this is a one-to-one function; this is not one-to-one.0628

They are both functions, since each element of the domain is paired only with one element of the range.0639

But in this case, it is not a unique element of the range: these two, -2 and -4, actually share an element of the range.0648

In other words, this is unique; it is a one-to-one correspondence.0656

OK, we can graph relations and functions by plotting the ordered pairs as points in the coordinate plane, as discussed a little while ago.0662

There are a couple of types of graphs that you can end up with.0673

The first is discreet, and the second is continuous; let's look at those two different types.0676

Consider this relation: OK, so if I am asked to graph this relation, I am going to graph each point:0682

(-4,-2): that is going to be right here; (-2,1)--right here; here, (0,2), 0 on the x, 2 on the y.0698

This is a discrete function--discrete graph--discrete relation.0714

This actually is both a relation and a function; so it is a discrete relation or a discrete function.0721

And the reason is because I have a set of discrete points; they are not connected.0727

And I can't connect them, because I haven't been given anything in between, or a way to know if or what lies in between these.0734

I can't just connect them when I don't know; there could be a point up here, or actually this is just the entire relation.0741

So, I can just work with what is given.0747

OK, a different scenario would be if I am given a relation y=x+1.0749

And I can go ahead and plot this out, if I say, "OK, when x is -1, -1+1 is 0; when x is 0, y is 1; when x is 1, 1+1 is 2; when x is 2, y is 3."0759

OK, so I am going to go ahead and plot this out.0779

When x is -1, y is 0; when x is 0, y is 1; when x is 1, y is 2.0781

Let's remove this out of the way.0790

When x is 1, y is 2; when x is 2, y is 3.0793

Now, I have a set of points, because these are the points I chose.0798

But because I am given this equation, there is an infinite number of points in between.0801

I could have chosen an x of .5 to get the value 1.5 here, to fill that in--and on and on, until this becomes continuous and forms a line.0807

So, this is a continuous function: the graph is a connected set of points,0821

so the relation (or the function in this case, since we do have a function) is continuous relation or continuous function,0834

because I have a line; whereas this, which is a set of points, is a discrete relation.0841

OK, one visual way to tell if a relation is a function is using the vertical line test.0849

And a relation is a function if and only if no vertical line intersects its graph at more than one point.0856

This is most easily understood through just working through an example.0866

Consider if you were given the following graph.0870

OK, the vertical line test: what you are seeing is, "Can you put a vertical line0874

somewhere on the graph so that it intersects the graph at more than one point?"0880

And I can: I put a vertical line here, and it intersects this graph at 1, 2, 3 places.0887

Over here, it only intersects at one place; that is fine; but if I can draw a vertical line anywhere on the graph0897

that intersects at more than one place, then we say that this failed the vertical line test.0903

And when something fails the vertical line test, it means that it is not a function.0912

The reason this works is that, if two or three or more points share the same x-value,0919

then they are going to lie directly above or below each other on the coordinate plane.0926

For example, looking right here, I have x = 3; x is 3; y is 0.0931

Then, I look right above it, up here: again, x is 3, and y is...say 2.1--pretty close.0940

Then, I look up here; again, x is 3, and y is about 4.6, approximately.0950

So, when x-values are the same, but then the y-values are different, that is telling me0959

that members of the domain are paired with more than one member of the range; by definition, that is not a function.0968

OK, consider a different graph--consider a graph like this of a line--a straight line.0975

OK, now, anywhere that I pass a vertical line through--anywhere on this graph--it is only going to intersect at one point.0984

So, this passed the vertical line test.0996

Therefore, this line, this graph, represents a function.1006

OK, so the vertical line test is a visual way of determining if a relation is a function.1014

Working with equations: an equation can represent either a relation or a function.1024

If an equation represents a function, then there is some terminology we use.1031

And let's start out by just looking at an equation that represents a function.1036

The variable corresponding to the domain is called the independent variable, and the other variable is the dependent variable.1043

So, here I have x and y; and let's look at some values--let's let x be -1.1050

Well, -1 times 2 is -2, minus 1--that is going to give me -3.1057

When x is 0, 0 times 2 is 0, minus 1 is -1.1063

When x is 1, 1 times 2 is 2, minus 1--y is 1.1071

When x is 2, 2 times 2 is 4, minus 1 gives me 3.1077

So, looking at how this worked, x is the independent variable.1084

The value of x is independent of y; I am just picking x's, and here it could be any real number.1099

We sometimes also say that this is the input; and the reason is that I pick a value for x (say 0),1109

and I put it in--I input it into the equation; then, I do my calculation, and out comes a y-value.1116

So, the value of y is dependent on x; therefore, it is the dependent variable; and we also sometimes say that it is the output.1127

You put x in and do the calculation; out comes the value of y; so x is independent, and y is dependent.1141

The notation that you will see frequently in algebra is function notation.1151

We have been writing functions like this: y = 4x + 3; but you will often see...1157

instead of an equation written like this, if it is a function, you will see it written as such.1165

And when we say this out loud, we pronounce it "f of x equals 4x plus 3."1170

And we are talking about the value of a function for a particular value of x.1179

So, we say, "The function of f at a particular x."1186

Let's let x equal 3; then, we can talk about f of 3--the value of the function, the value of y,1193

of the dependent variable, when the independent variable, x, is 3.1204

And in that case, since it is telling us that x is 3, I am going to substitute in 3 wherever there is an x.1209

And I could calculate that out to tell me that f(3) is...4 times 3 is 12, plus 3...so f(3) is 15.1217

Here, x is an element of the domain, of the independent variable; f(x) is an element of the range.1231

So again, we are going to be using this function notation throughout the remainder of the course.1246

Looking at the first example: the relation R is given by this set of coordinate pairs.1253

Give the domain and range, and determine of R is a function.1260

Well, recall that the domain is comprised of the first element of each of these coordinate pairs.1267

So in this case, the domain would be {1, 2, 6, 5, 7}.1275

The range: the range is comprised of the second element of each ordered pairs, so I have 4, 3...4 again;1291

I don't need to write that again; 3--I have 3 already; and 5; I am just writing down the unique elements.1303

This is the domain, and this is the range.1309

Now, is R a function? Well, I can always use mapping to just help me determine that.1311

And I am going to write down my members of the domain, and my elements of the range.1320

And then, I am going to use arrows to show the correspondence between each:1330

1 and 4--1 is paired with 4; 2 is paired with 3; 6 is also paired with 4; 5 is paired with 3; and 7 is paired with 5.1333

Now, I am looking, and I only see one arrow leading from each element of the domain.1350

There is no element of the domain that is paired with two elements of the range.1354

So, in this case, this is a function; so, is R a function? Yes, this relation is a function--R is a function.1359

So, always double-check and make sure you have answered each part.1373

I found the domain; I found the range; and I determined that R is a function.1375

OK, the relation R is given by the equation y=2x2+4; is R a function?1384

What are the domain and range? Is R discrete or continuous?1393

Let's just look at some values for x and y to help us determine if this relation is a function.1400

If I let x equal -1, -1 times -1 is 1, times 2 is 2, plus 4 is 6.1412

OK, if x is 0, this is 0, plus 4--that gives me 4.1422

If x is 3, 3 squared is 9, times 2 is 18, plus 4 is 22.1428

So, as you are going along, you can see that, for any value of x, there is only one value of y; therefore, R is a function.1434

What is the domain? Well, I could pick any real number for an x-value that I wanted, so the domain is all real numbers.1450

You might, at first glance, say, "Oh, the range is all real numbers, as well"; but that is not correct, because look at what happens.1467

Because this is x2, whenever I have a negative number, it becomes positive; if I have a positive number, it stays positive, of course.1475

Therefore, if I have, say, -1, that becomes 1; this becomes 6.1486

So, I am not going to get any value lower than...for y, the smallest value I will get is for when x is 0.1493

OK, so if x is 0, y is 4; because -1 is going to give me a bigger value--it is going to give me 6.1505

If I do -2, that is going to be 4 times 2 is 8, plus 4 is 12.1510

So, the lowest value that I will be able to get for y will occur when x is 0.1516

And that is going to give me a y-value of 4.1522

Therefore, the range is that y is greater than or equal to 4.1525

So, the most difficult part of this was just realizing that the range is not as broad as it looked initially.1530

Because this involves squaring a number, there is a limit on how low you are going to go with the y-value.1538

So, this is a range with a domain of all real numbers, and a range of greater than or equal to 4.1545

OK, in Example 3, graph the relation R given by 2x - 4y = 8.1559

Is R a function? Find its domain and range. Is R discrete or continuous?1572

OK, so graph the relation given by 2x - 4y = 8.1583

Let's go ahead and find some x and y values, so that we can graph this.1590

When x is 0, we need to be able to solve for y; when x is 0, let's figure out what y is.1598

0 - 4y equals 8; therefore, y equals -2 (dividing both sides by -4).1607

OK, when x is 2, 2 times 2 minus 4y equals 8; that is 4 minus 4y equals 8; that is -4y equals 4; y = -1.1615

And let's do one more: when x is -2, this is going to give me -4 - 4y = 8.1635

That is going to then give me, adding 4 to both sides, -4y = 12, or y = -3; that is good.1646

All right, so when x is 0, y is -2; when x is 2, y is -1; when x is -2, y is -3, right here.1657

I am asked to graph it; and I have some points here that I generated,1686

but I also realize that I could have picked points in between these, which would actually end up connecting this as a line.1691

So, I am not just given a set of ordered pairs; I am given an equation that could have an infinite number of values for x,1702

which would allow me to graph this as a continuous line.1707

Therefore, I graphed the relation...is R a function?1713

Is R discrete or continuous? Well, I have already answered that--seeing the graph of this, I know that this is continuous.1720

And let's see, the next step: is R a function?1731

Yes, it is a function, because if I look, for every value of x (for every value of the domain), there is one value only of the range.1746

So, every element of the domain is paired with only one element of the range.1758

It is continuous, and it is a function.1762

Find the domain and range: well, this is another case where I could choose x to be any real number, so it would be all real numbers--any real number.1765

Here, the situation is the same for the range--all real numbers.1778

Depending on my x-value, I could come up with infinite possibilities for what the range would be, what the y-value would be.1784

R is a function; its domain and range are all real numbers; and this is a continuous function.1792

OK, in Example 4, we are given f(x) = 3x2 - 4, and asked to find f(2), f(6), and f(2k).1805

First, f(2): recall that, when you are asked to find a function for a particular value of x,1824

you simply substitute that value for x in the equation; so f(2) equals 3(4) - 4, so that is 12 - 4; so f(2) = 8.1834

Next, I am asked to find f(6), and that is going to equal 3(62) - 4.1852

f(6) = 3(36) -4, and that turns out to be 108 - 4, so f(6) is 104.1860

Now, at first, this f(2k) might look kind of difficult; but you treat it just the same as you did with the numbers, when x is a numerical value.1877

Everywhere I see an x, I am going to insert 2k.1887

And figuring this out, 2 times 2, 2 squared, is 4; k times k is k2.1891

3 times 4 is 12, so I have 12k2 - 4; so f(2k) = 12k2 - 4.1902

So again, if you are asked to find the function of a particular value of x, you simply substitute whatever is given, including variables, for x.1910

That concludes this lesson of Educator.com; I will see you back here soon!1919

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