INSTRUCTORS Carleen Eaton Grant Fraser

Dr. Carleen Eaton

Special 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 Julia Lee on July 7, 2020Can you further explain floor functions, they were in the practice questions. 0 answersPost by Munqiz Minhas on September 7, 2016You should include economic applications problems (profit functions)! I have a test on this and struggling a little. 1 answerLast reply by: Dr Carleen EatonThu Jul 31, 2014 6:58 PMPost by Philippe Tremblay on July 17, 2014At 23:40, the 'open circle' should be face to 3 (value of y). 0 answersPost by Kavita Agrawal on May 20, 2013Example 1 is a step function, not an absolute value function as you labelled in the side bar. 0 answersPost by Tearion Scott on June 25, 2011Why don't you include application of the problems related to economics or business? Still you do a great job. :-)

### Special Functions

• Step functions have graphs that are a series of steps. The steps can be going upward or downward.
• Constant functions are of the form f(x) = c, where c is a real number. Their graphs are horizontal lines.
• A piecewise function is defined differently on different intervals in the domain. The graph consists of a different graph on each interval
• The absolute value function is an example of a piecewise function. Its graph looks like a V, translated horizontally or vertically.

### Special Functions

Graph f(x) = [x]
• Recall that this function is called the Greatest Integer Function. Another appropriate name for this function is the floor function because whatever value you get inside, it's brought to the floor, or to the number less than or equal to that whole integer.
• Modern computer algebra systems have a built - in function g(x) = floor(x + 2) to allow you to work with these types of functions.
• The greatest integer that is less than or equal to 4.7
• Complete the table below, remember, you're looking for the floor of every number inside the [  ].
•  x f(x) = [x] -2.9 -2.1 -2 -1.9 -1.1 -1 0 0.9 1 1.1 1.9 2.1 2.9 3 3.1 3.9 4.0 4.1
•  x f(x) = [x] -2.9 [-2.9]= floor(-2.9) = -3 -2.1 [-2.1]= floor(-2.1) = -3 -2 [-2]= floor(-2) = -2 -1.9 [-1.9]= floor(-1.9) = -2 -1.1 [-1.1]= floor(-1.1) = -2 -1 [-1]= floor(-1) = -1 0 [0]= floor(0) = 0 0.9 [0.9]= floor(0.9) = 0 1 [1]= floor(1) = 1 1.1 [1.1]= floor(1.1) = 1 1.9 [1.9]= floor(1.9) = 1 2.1 [2.1]= floor(2.1) = 2 2.9 [2.9]= floor(2.9) = 2 3 [3]= floor(3) = 3 3.1 [3.1]= floor(3.1) = 3 3.9 [3.9]= floor(3..9) = 3 4.0 [4.0]= floor(4) = 4 4.1 [4.1]= floor(4.1) = 4
• Graph it and follow the convention shown in lecture. This can't be shown in the graph provided
Graph f(x) = [x + 3]
• Recall that this function is called the Greatest Integer Function. Another appropriate name
• for this function is the floor function because whatever value you get inside, it's brought
• to the floor, or to the number less than or equal to that whole integer. Modern computer algebra systems
• have a built - in function g(x) = floor(x + 2) to allow you to work with these types of functions.
• The greatest integer that is less than or equal to 4.7
• Complete the table below, remember, you're looking for the floor of every number inside the [ ].
•  x f(x) = [x+3] -2.9 -2.1 -2 -1.9 -1.1 -1 0 0.9 1 1.1 1.9 2.1 2.9 3 3.1 3.9 4.0 4.1
•  x f(x) = [x+3] -2.9 [-2.9+3]= floor(0.1) = 0 -2.1 [-2.1+3]= floor(0.9) = 0 -2 [-2+3]= floor(1) = 1 -1.9 [-1.9+3]= floor(1.1) = 1 -1.1 [-1.1+3]= floor(1.9) = 1 -1 [-1+3]= floor(2) = 2 0 [0+3]= floor(3) = 3 0.9 [0.9+3]= floor(3.9) = 3 1 [1+3]= floor(4)= 4 1.1 [1.1+3]= floor(4.1) = 4 1.9 [1.9+3]= floor(4.9) = 4 2.1 [2.1+3]= floor(5.1) = 5 2.9 [2.9+3]= floor(5.9) = 5 3 [3+3]= floor(6) = 6 3.1 [3.1+3]= floor(6.1) = 6 3.9 [3.9+3]= floor(6.9) = 6 4.0 [4.0+3]= floor(7) = 7 4.1 [4.1+3]= floor(7.1) = 7
• Graph it and follow the convention shown in lecture. This can't be shown in the graph provided
Graph f(x) = |x| + 1
• This is the graph of the absolute value function of x.
• Remember that the absolute valure refers to the distance away from zero, eg. | − 4| = 4,| − 6| = 6
• Complete the table of values below and graph
•  x f(x) = |x| + 1 -3 -2 -1 0 1 2 3 4
•  x f(x) = |x| + 1 -3 |−3| + 1 = 3 + 1 = 4 -2 |−2| + 1 = 2 + 1 = 3 -1 |−1| + 1 = 1 + 1 = 2 0 |0| + 1 = 0 + 1 = 1 1 |1| + 1 = 1 + 1 = 2 2 |2| + 1 = 2 + 1 = 3 3 |3| + 1 = 3 + 1 = 4 4 |4| + 1 = 4 + 1 = 5
• Graph it.
Graph f(x) = − |x| + 1
• This is the graph of the absolute value function of x.
• Remember that the absolute valure refers to the distance away from zero, eg. | − 4| = 4,| − 6| = 6
• Complete the table of values below and graph
•  x f(x) = −|x| + 1 -3 -2 -1 0 1 2 3 4
•  x f(x) = −|x| + 1 -3 −|−3| + 1 = − 3 + 1 = −2 -2 −|−2| + 1 = −2 + 1 = −1 -1 −|−1| + 1 = −1 + 1 = 0 0 −|0| + 1 = −0 + 1 = 1 1 −|1| + 1 = −1 + 1 = 0 2 −|2| + 1 = −2 + 1 = −1 3 −|3| + 1 = −3 + 1 = −2 4 −|4| + 1 = −4 + 1 = −3
• Graph it
Graph f(x) = |x + 3| + 1
• This is the graph of the absolute value function of x.
• Remember that the absolute valure refers to the distance away from zero, eg. | − 4| = 4,| − 6| = 6
• Complete the table of values below and graph
•  x f(x) = |x+3| + 1 -6 | − 6 + 3| + 1 = 3 + 1 = 4 -5 | − 5 + 3| + 1 = 2 + 1 = 3 -4 | − 4 + 3| + 1 = 1 + 1 = 2 -3 | − 3 + 3| + 1 = 0 + 1 = 1 -2 | − 2 + 3| + 1 = 1 + 1 = 2 -1 | − 1 + 3| + 1 = 2 + 1 = 3 0 |0 + 3| + 1 = 3 + 1 = 4 1 |1 + 3| + 1 = 4 + 1 = 5
•  x f(x) = |x+3| + 1 -6 | − 6 + 3| + 1 = 3 + 1 = 4 -5 | − 5 + 3| + 1 = 2 + 1 = 3 -4 | − 4 + 3| + 1 = 1 + 1 = 2 -3 | − 3 + 3| + 1 = 0 + 1 = 1 -2 | − 2 + 3| + 1 = 1 + 1 = 2 -1 | − 1 + 3| + 1 = 2 + 1 = 3 0 |0 + 3| + 1 = 3 + 1 = 4 1 |1 + 3| + 1 = 4 + 1 = 5
• Graph it.
Graph f(x) = |x − 3| + 1
• This is the graph of the absolute value function of x.
• Remember that the absolute valure refers to the distance away from zero, eg. | − 4| = 4,| − 6| = 6
• Complete the table of values below and graph
•  x f(x) = |x−3| + 1 -1 0 1 2 3 4 5 6
•  x f(x) = |x−3| + 1 -1 | − 1 − 3| + 1 = 4 + 1 = 5 0 |0 − 3| + 1 = 3 + 1 = 4 1 |1 − 3| + 1 = 2 + 1 = 3 2 |2 − 3| + 1 = 1 + 1 = 2 3 |3 − 3| + 1 = 0 + 1 = 1 4 |4 − 3| + 1 = 1 + 1 = 2 5 |5 − 3| + 1 = 2 + 1 = 3 6 |6 − 3| + 1 = 3 + 1 = 4
• Graph it.
Graph f(x) = x2  if  0 ≤ x ≤ 3x − 3  if  x < 0
• There are two different pieces/sections in the graph, the function is defined differently for different intervals in the domain.
• Whenever you see a > or < sign in the domain, you my still calculate the value, however, when graphed, it must be represented with an open circle.
• Complete the table below and graph the Piece - wise Function.
• Section 1:
Domain: 0 ≤ x ≤ 3

 x f(x) = x2 0* 1 2 3*

*must be closed circle
• Section 1:
Domain: 0 ≤ x ≤ 3

 x f(x) = x2 0* (0)2 = 0 1 (1)2 = 1 2 ( 2)2 = 4 3* ( 3 )2 = 9

*must be closed circle
• Section 2:
Domain: x < 0

 x f(x) = x−3 0* 1 2 3

*must be closed circle
• Section 2:
Domain: x < 0

 x f(x) = x-3 0* 0-3=-3 1 -1-3=-4 2 -2-3=-5 3 -3-3=-6

*must be closed circle
• Graph the piece - wise function. Pay close attention to open and close circles on the graph.
Graph f(x) = [3/4]x−1  if  4 ≤ x < 8 −[5/4]x+3  if  0 < x < 4
• There are two different pieces/sections in the graph, the function is defined differently for different intervals in the domain.
• Whenever you see a > or < sign in the domain, you my still calculate the value, however, when graphed, it must be represented with an open circle.
• Complete the table below and graph the Piece - wise Function.
• Notice how both sections are linear functions, therefore, you may find the end - points of the intervals of the domain only. Play close attention to open or closed circles.
• Section 1:
Domain: 4 ≤ x < 8

 x f(x) =[3/4]−1 4 (Closed Circle) 8 (Open Circle)
• Section 1:
Domain: 4 ≤ x < 8

 x f(x) =[3/4]−1 4 (Closed Circle) [3/4](4)−1=3−1=2 8 (Open Circle) [3/4](8)−1=6−1=5
• Section 2:
Domain: 0 < x < 4

 x f(x) =−[5/4]+3 0 (Open Circle) 4 (Open Circle)
• Section 2:
Domain: 0 < x < 4

 x f(x) =−[5/4]+3 0 (Open Circle) −[5/4](0)+3=3 4 (Open Circle) −[5/4](4)+3=−5+3=−2
Graph f(x) = x2−1  if  −3 ≤ x < 1 3x+1  if  1 < x ≤ 5
• There are two different pieces/sections in the graph, the function is defined differently for different intervals in the domain.
• Whenever you see a > or < sign in the domain, you my still calculate the value, however, when graphed, it must be represented with an open circle.
• Complete the table below and graph the Piece - wise Function.
• Notice how the second sections is a linear function, therefore, you may find the end - points of the interval of the domain only. Play close attention to open or closed circles.
• Section 1
Domain: −3 ≤ x < 1

 x f(x) =x2−1 -3 (Closed Circle) -2 -1 0 1 (Open Circle)
• Section 1
Domain: −3 ≤ x < 1

 x f(x) =x2−1 -3 (Closed Circle) (−3)2−1=9−1=8 -2 (−2)2−1=4−1=3 -1 (−1)2−1=1−1=0 0 (0)2−1=0−1=−1 1 (Open Circle) (1)2−1=1−1=0
• Section 2
Domain: 0 < x ≤ 5

 x f(x) = 3x+1 1 (Open Circle) 5 (Closed Circle)
• Section 2
Domain: 0 < x ≤ 5

 x f(x) = 3x+1 1 (Open Circle) 3(1)+1=4 5 (Closed Circle) 3(5)+1=16
• Graph the piece - wise function. Pay close attention to open and close circles on the graph.

*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.

### Special 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
• Step Functions 0:07
• Example: Apple Prices
• Absolute Value Function 4:55
• Example: Absolute Value
• Piecewise Functions 9:08
• Example: Piecewise
• 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

### Transcription: Special Functions

Welcome to Educator.com.0000

In today's session, we are going to talk about several special functions.0002

The first one we are going to discuss is the step function; and the step function makes a unique-looking graph.0007

It is a function that is constant for different intervals of real numbers.0013

And the result is a graph that is a series of horizontal line segments, so they look like steps; and that is where the name "step function" comes from.0018

The best way to understand this is through an example.0028

So, for example, if apples were sold at a price of a dollar per pound, and the price is such that0030

you are charged $1 for each pound, or any part of a pound--in other words, they round up in order to determine the price;0053 if you had a pound and a half of apples, they are going to charge you$1 for the pound, and then another $1 for the half pound.0070 So, they charged you a dollar for a pound, and then any part of a pound is considered a full pound in terms of pricing.0079 So, if x is the number of pounds, and y is the cost, then let's see what kind of values we get and what the graph looks like.0088 OK, if I have .8 pounds, I am going to get charged$1: they are going to round up.0104

If I have a full pound, I am going to get charged $1; if I have a little bit over a pound--I have 1.2 pounds--0113 I am going to get charged$1 for the first pound, and then that .2 is going to be another dollar, so $2.0124 1.4 pounds--again,$2, and on up...2 pounds--a dollar for the first pound and a dollar for the second pound.0135

2.5--$2, and then the .5 is another$1, so that bumps me up to $3; until I hit 3 pounds, it is$3, also.0145

3.8: $1 for the first pound,$1 for the second pound, $1 for the third pound, and another$1 for that .8, so $4.0155 So, you can see that the function is constant for different intervals.0166 So, for this first interval, from a little bit above 0, all the way to 1, the y-value is constant; it is$1.0170

For the next interval, which is just above 1, all the way to 2, including 2, it is going to be $2.0180 Once I get above 2, up to and including 3, it is$3; and so on.0188

So, this is constant for different intervals of real numbers.0193

Looking at what the graph looks like: any part of a pound up to and including a dollar for that first pound is a dollar.0198

Now, 0 pounds of apples are going to be $0; so I am not going to include 0.0207 But even .1...the slightest part of a pound is going to be$1, so I put an open circle to indicate that 0 is not included;0213

but just above that is, all the way up to and including 1; since 1 is included (1 is also a dollar), I am going to put that as a closed circle.0224

So here, I have pounds; and here, I have the cost on the y-axis in dollars.0233

Now, once I get just above 1 (say 1.1 pounds), they are going to charge me $2--not including 1, but just above it--open circle.0243 All the way up to 2...at 2 pounds, I will also be charged$2.0254

Once I hit just above 2, I am going to be charged \$3, all the way to 3 and including 3.0260

And on...so, you can see how this looks like a series of steps, and how this is a result of the fact that the function is constant for different intervals of x.0270

This function is the same for this entire interval; then it is the same for the second interval; and on.0287

So, this is a step function.0292

A second type of function that you will be working with is an absolute value function.0295

And these functions have special properties: looking first at f(x) = |x|, just the simplest case, here f(x) is just going to be the absolute value of x.0301

When x is 0, f(x) is also 0; when x is 1, the absolute value is 1; and so on for positive numbers.0317

Now, let's look at negative numbers: for -1, the absolute value is 1; -2--the absolute value of x is 2; and on.0329

The result is a certain shape of graph: when x is 0, f(x) is 0; x is 1, f(x) is 1; x is 2, f(x) is 2; and on.0343

Now, for negative numbers: -1, f(x) is 1; -2, f(x) is 2; -3, f(x) is 3; and it is going to continue on like that.0358

So, absolute value graphs are v-shaped; so we are going to end up with a v-shaped graph.0372

Depending on the function, the graph can be shifted up, or it can be shifted to the right or to the left; and let's see how that could happen.0385

Let's now let f(x) equal (here f(x) equals |x|)...let's say f(x) equals the absolute value of x, plus 1.0393

OK, so we are given x, and the absolute value of x is 0; we are adding 1 to them, so this is going to become 1.0404

The absolute value of x is 1; 1 + 1 is 2; 3; 4; the absolute value of -1 is 1; 1 + 1 is 2.0412

The absolute value of -2 is 2; add one to that--it is 3; and add 1 to 3 to get 4.0425

OK, so it is the same as this, except increased by 1: each value of the function has been increased by 1.0432

So now, let's see what my graph is going to look like.0439

Right here, I have the graph for f(x) = |x|; now, I am going to look at this graph.0443

When x is 0, f(x) is 1; when x is 1, f(x) is 2; when x is 2, f(x) is 3; so you can see what is happening.0451

And then here, I have x is 3, f(x) is 4; negative values--when x is -1, f(x) is 2; when x is -2, f(x) is 3; when x is -3, f(x) is 4.0465

OK, I am drawing the line through this: this is the graph of f(x) = |x| + 1.0492

So you see that this graph, the v-shaped graph, is simply shifted up by 1.0509

And again, you can also shift this from side to side; and we will see an example of that later on.0513

So, in the absolute value function, it is very important to find both negative and positive values of the function.0519

So, assign x 0; assign it some positive values; and it is very important to find what f(x) will be when x is negative,0526

because if I didn't--if I picked only positive values--I would end up with half of a graph.0537

So, to get the entire v-shape, choose negative and positive values for x.0541

The third special function that we are going to discuss is called a piecewise function.0548

And a piecewise function is a function that is described using two or more different expressions.0553

The result is a graph that consists of two or more pieces.0560

Just starting out with one that consists of two pieces: the notation is usually like this--one large brace on the left:0564

f(x) equals x + 2 for values of x that are less than 3; and f(x) equals 2x - 3 for values of x that are greater than or equal to 3.0573

So you see that, for different intervals of the domain, the function is defined differently.0590

So, it is a function that is described using two or more different expressions.0597

So, for the part of the domain where x is less than 3, this is the function.0600

For the interval of the domain that is greater than or equal to 3, this is the function that I am going to use.0605

Let's see what happens: let's first use this part of the function where f(x), or y, equals x + 2.0613

And for the domain, remember that x is going to be less than 3.0625

So, I will go ahead and start out with 2: when x is 2, f(x) (or y) is 4.0630

When x is 1 (I have to remain at x-values less than 3), f(x) is 3; when x is 0, f(x) is 2.0636

Just picking a negative number: when x is -4, f(x) is -2.0648

I am going to graph that here; OK, when x is 2, f(x), or y, is 4; x is 1, y is 3; x is 0, y is 2; x is -4, y is -2.0656

OK, now, this is for values of x that are less than 3; 3 is about here.0689

Therefore, anything just below 3, but not including 3, is going to be part of this graph.0698

So, we are going to use an open circle here to indicate that 3 is not going to be included as part of this function--the domain of this function.0703

So, it is going to begin at just below 3 and continue on indefinitely; that is the first piece of the graph.0716

The second piece of the graph is for x such that x is greater than or equal to 3; and here, y is going to be 2x - 3.0723

So, it is greater than or equal to 3, so I am first going to let x equal 3.0735

3 times 2 is 6, minus 3 is 3; getting larger--when x is 4, it is 2 times 4 is 8, minus 3 is 5; when x is 5, 5 times 2 is 10, minus 3 is 7.0738

Now, this does include 3--this section of the graph--this piece; so, when x is 3, y is 3, right here.0755

When x is 4, y is 5, right here; when x is 5 (let me shift that over just a bit), y is 7; OK.0768

Now, if you look, this actually did end up including all possible values of x (all real numbers),0801

because when x is less than 3, I use this function; and then, as soon as x becomes 3 or greater, I shift to this other function.0807

So, you can see how there are two pieces to the graph; and you actually can have situations where there are more than 2.0816

You could be given, say, f(x) is 4x + 3, 2x + 7, and x - 1, and then given limits on the domain for each of those.0822

So, there are at least two pieces; however, there can be more.0836

In Example 1, we have a greatest integer function.0840

Before we start working in this, let's just review what we mean by the greatest integer function.0845

So, when you see this notation with the brackets, let's say that you have a number in here, such as 4.7.0848

What this is saying is that this value is equal to the greatest integer that is less than or equal to 4.7; so that is 4.0855

It is the greatest integer less than or equal to whatever is in here; if it was 2.8, it would be 2.0866

Be careful with negative numbers: let's say I have -3.2--the temptation is to say, "Oh, that is equal to -3";0875

but if you look at it on the number line, -3.2 is right about here; OK, so if I have -3.2,0882

and I am trying to find the greatest integer that is less than or equal to 3.2, it is going to have to be something over here--smaller.0893

So, it is actually going to be -4; so just be careful when you are working with negative numbers.0903

Whatever is inside here--whatever that value is--the function is equal to the greatest integer that is less than or equal to this value in here.0907

Understanding that, you can then find the graph; so let's find a bunch of points for this,0920

so we make sure we know what is going to happen with various situations.0927

When x is 0, this inside here is going to be 2; the greatest integer less than or equal to 2 is 2.0935

When x is .6, then you are going to get 2.6 in here; the greatest integer less than or equal to 2.6 is 2.0945

.8--I get 2.8; again, I round down to 2.0953

All right, so when I hit 1, 1 plus 2 is 3, and the greatest integer less than or equal to 3 is 3.0958

Slightly above 1: that is going to give me 1.2 + 2 is 3.2; the greatest integer less than or equal to 3.2 is also 3.0967

OK, so you can get the idea of what this is going to look like.0976

And that continues on; and then, when we hit 2, 2 + 2 is 4; the greatest integer is going to be 4.0980

For negative numbers: let's take -.5: -.5 and 2 is 1.5; the greatest integer less than or equal to 1.5 is going to be 1.0988

Now, notice: I have a negative number for x, but this did not come out to be a negative number; so that is different from the case I was discussing there.1005

Let's go a little bit bigger--let's say -3: -3 and 2 is -1, and that is going to be -1.1014

Let's say I take -3.5: -3.5 and 2 is going to equal -1.5: again, just thinking about that to make sure you have it straight,1024

I have -1.5; so I have 0; I have -1; I have -1.5; I have -2; the greatest integer less than or equal to this is actually -2.1037

OK, now plotting this out: when x is 0, f(x) is 2; when x is slightly above 0 (it's .6), f(x) is 2; .8--it is 2, all the way up until I hit 1.1053

At 1, f(x) becomes 3; therefore, 1 is not included in this interval.1075

So, you can already see that this is going to be a step function, because we have intervals.1082

For different intervals of the domain, we have that same value for the range.1089

All right, for values between 1 and 2, f(x) will be 3; once we hit 2, I have to do an open circle, because at 2, the value for f(x) jumps up to 4.1095

OK, so you can see what this is going to look like; and that pattern is just going to continue.1112

Let's look over here at negative numbers: when x is slightly less than 0, then you are going to end up with an f(x) that is 1.1116

So, for values slightly less than 0, but not including 0, this is what you are going to end up with.1137

OK, looking, say, when x is -3: when x is -3, f(x) will be -1.1145

But when we go slightly more negative than that, when x is -3.5, f(x) is going to be -2; it is going to be down here.1161

So, the steps on this side are going to have the open circle on the right.1175

And I am going to jump down, and it is not going to include -2, because -2 and 2 is 0;1184

so -2 is going to be right here for the x-value, and the f(x) will be 0.1193

But as soon as I get to a little bit bigger than -2, the greatest integer is going to be down here.1199

OK, and so, we continue on like that with the steps; and you can see how this is a step function.1207

You just have to be very careful and pick multiple points until you can see the pattern1220

where for a certain interval of the domain, the range is a particular value.1225

OK, so that was a step function, and it involved the greatest integer function.1234

Example 2: now we are working with absolute value.1240

g(x) equals the absolute value of x, minus 3.1244

And we already know that the shape of this graph is going to be in a v.1248

But we don't know exactly where that v is going to land, so let's plot it out.1252

When x is 0, the absolute value of x is 0; minus 3--that gives me -3.1258

When x is 1, the absolute value is 1; minus 3...g(x) is -2.1264

When x is 2, the absolute value is 2; minus 3 is going to give me -1.1269

Now, let's pick some negative numbers for x, because that is really important to do with an absolute value graph.1277

When x is -1, the absolute value is 1, minus 3 gives me -2; you can already see that my v shape is going to occur.1283

When x is -2, the absolute value is 2; minus 3 is -1.1291

The absolute value of -3 is 3; minus 3 is 0; so this is enough to go ahead and plot.1299

x is 0; g(x) is -3; x is 1, g(x) is -2; x is 2, g(x) is -1; over here with the negative values,1305

when x is -1, g(x) is -2; when x is -2, g(x) is -3; when x is -3, g(x) is 0.1316

So, you can see that I have a v-shaped graph, and compared with my graph that would look like this,1325

that would have the v starting right here, it has actually shifted down by 3; that is an absolute value function.1338

Here you can see that you are given a piecewise function, because there are two different pieces.1348

And this could also be written in this notation.1357

There are two different sections to the graph; and we see that the function is defined differently for different intervals of the domain.1360

Starting with if x is greater than 2 (this is going to be for x-values where x is greater than 2): f(x) is going to be x + 1.1371

When x is 3, f(x) is 4; when x is 4, f(x) is 5; when x is 5, f(x) is 6; OK.1386

When x is 3, f(x) is 4; when x is 4, f(x) is 5; and it is going to go on up.1404

And that is going to go all the way, until just greater than 2.1416

2 is not going to be included on this graph, because it is a strict inequality; so I am going to use an open circle, and this is going to continue on.1422

Now, for x less than or equal to 2, I have a different situation: I am looking at f(x) is -2x.1430

OK, so when x is 2, 2 time -2 is -4; when x is 1, 1 times -2 is -2; when x is 0, f(x) is 0; when x is -2, that is -2 times -2, which is positive 4.1443

So, starting with x is 2: when x is 2, f(x) is -4; and that is including the 2.1464

When x is 1 (these are values less than or equal to 2, so I am getting smaller), f(x) is -2.1476

x is 0; f(x) is also 0; when x is -2, f(x) is up here at 4; OK, so I have a steep line going right through here.1494

So, you can see: this is a piecewise function consisting of two pieces; and here, one picks up where the other leaves off.1509

For values greater than 2, this is my graph; for values of x less than or equal to 2, this is my graph; so this is a piecewise function.1515

OK, this time, in Example 4, we have both greatest integer and absolute value in this function.1525

Recall that, for the greatest integer function, what that is saying is that whatever is inside this bracket--let's say it's 1.2--1532

it is asking for the greatest integer less than or equal to 1.2; in that case, this would be 1.1540

Or if I had 4.8, it would be 4.1545

For negative numbers, like -3.2, the greatest integer less than or equal to -3.2 is -4.1551

OK, now, since this is a bit complicated, it is helpful just to take it in stages.1560

So, I am going to look first at what the greatest integer of x is; and then, I am going to look for the absolute value of what that is.1567

If x is .2, the greatest integer less than or equal to .2 is 0; the absolute value of 0 is 0; so this is the function that we are looking for.1578

And the same would hold true of .5: round down to 0; the absolute value is 0.1590

When we hit 1, the greatest integer less than or equal to 1 is 1, and the absolute value of that is 1.1596

1.2: again, we are going to go down to the greatest integer that is less than or equal to 1.2, which is 1; and the absolute value is 1.1608

The same for 1.8, and all the way up until 2; once we hit 2, the greatest integer less than or equal to 2 is 2; the absolute value is 2.1621

So, that is working with positive numbers, greatest integer, and the absolute value; it is the same; OK.1629

So, let's go to negative: for -.4, the greatest integer that is less than or equal to -.4...I am looking, and I have 0, and 1,1636

and -1, and -.4 is about here; so I am going to go down to -1; the absolute value of that is 1.1650

Here you can see that the greatest integer is not the same as the absolute value.1659

Or for -1, the greatest integer less than or equal to -1 is -1; the absolute value is 1.1664

-1.8: the greatest integer that is less than or equal to -1.8...I am going to go down to -2; and the absolute value is 2.1674

For -2, the greatest integer less than or equal to -2 is -2; the absolute value is 2.1686

So, you see that there are intervals here--intervals of the domain end up with the same value for the function.1694

So, I am going to have a step function.1704

But remember that absolute value graphs are v-shaped, so I am going to end up with a v-shaped step function.1709

Let's plot these out: for 0, the greatest integer of 0 would be 0, and then the absolute value would be 0.1715

So, with 0, we are going to include it; and for all values up to but not including 1, the function is going to equal 0.1726

Once we get to 1, I have an open circle, because it is not included.1736

When x is 1, f(x) is 1; so I am going to jump up here.1740

All the values between 1 and 2, but not including 2, will have an f(x), or a y-value, that is 1.1746

As soon as I hit 2, open circle: I am going to jump up, and once I hit 2, f(x) is 2, all the way up to, but not including, 3.1754

And it is going to go on that way: and you see now, we have the step function, and it is v-shaped like absolute value.1771

Let's look over here on the negative side of things.1778

For -.4, somewhere in here, it is going to equal 1; -1 is also equal to 1; so here, on the left side, I have a closed circle, and an open circle on the right.1782

It is the opposite of what I had over here.1799

When I get to less than -1, my value for f(x) is going to jump up to 2; this is a closed circle;1802

I get slightly less than, but not including, -1; it is going to jump up to 2.1815

-2: my value is also 2, and everything in between; and then, when I get to just slightly more negative than -2, like -2.1, it is going to jump up to 3.1822

You can see how this is v-shaped, and it is a step function.1841

The step function comes from it being the greatest integer function; the v shape comes from that absolute value.1847

And you also just had to be careful how you are doing the open and the closed circles; OK.1853

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