Piecewise-Defined Functions
Real life example
Evaluate a Piecewise Function
Graph
Domain and Range
Evaluate a Piecewise Function
Evaluate
Find
Graph Piecewise Functions
Example
Cut the piece
Graph the function
Graph Piecewise Functions
Graph the function
Cut the piece
Graph Piecewise Functions
Domain and Range of Functions Graph
Example:
>
Range
扫描全能王 创建
扫描全能王 创建
Domain and Range of Functions
Ordered Pairs
Mapping
Graph
Equation
Domain and Range of Functions
Ordered Pairs
Domain
Range
Domain and Range of Functions
Mapping
1
2
3
Domain
Range
A
B
4
Domain and Range of Functions Graph
>
>
Domain
Range
>
>
Domain
Range
>
>
Domain
Range
>
>
Domain
Range
Domain and Range of Functions Graph
>
>
Domain
Range
>
>
>
Domain
Range
>
>
>
>
Domain
Range
>
>
>
Domain
Range
>
Domain and Range of Functions Graph
Domain
Range
Example
Domain and Range of Functions Graph
>
Domain
Range
Domain
Range
>
Domain
Range
Examples
Domain
Range
Domain and Range of Functions
Equation
Steps to find the domain of a function defined by an equation
1.- Start with all real numbers,
2.- Exclude zeros from the denominator, if any.
3.- Exclude negative numbers from even index radicals, if any.
What is ?
Find the domain of
For which values of is the denominator zero?
For
Exclude zero from
Set the radicand of an even index root to be
Domain and Range of Functions Equation
Examples:
Domain of is
1)
2)
For which values of is
1.- Start with all real numbers,
2.- Exclude zeros from the denominator, if any.
3.- Exclude negative numbers from even index radicals, if any.
Set the radicand of an even index root to be
+5
+5
Domain of is
Examples
Domain and Range of Functions Equation
Examples:
3)
1.- Start with all real numbers,
2.- Exclude zeros from the denominator, if any.
3.- Exclude negative numbers from even index radicals, if any.
Set the radicand of an even index root to be
Domain of is
Which values of make the denominator zero?
and
Exclude from
Examples
or
Domain and Range of Functions Equation
Examples:
4)
1.- Start with all real numbers,
2.- Exclude zeros from the denominator, if any.
3.- Exclude negative numbers from even index radicals, if any.
Set the radicand of an even index root to be
Domain of is
5)
-4
-4
-3
-3
Domain of is or
Must be positive
And not zero
Examples
Domain and Range of Functions Equation
4)
and
and
-12
-12
3
3
and
-4
0
5
[
Exclude
Domain of is or
Examples
Functions
Relations
Definition of Functions
Function Notation
The Graph of a Function
Relations
A Relation is a set of ordered pairs
It may be specified in 4 ways
By a Graph
By mapping
By displaying the pairs
By an equation
Domain
Range
1
2
3
a
b
c
Domain
Range
Domain
Range
Input
Output
Domain
Range
Range
Domain
.
You may also use tables
| 1 | b |
| 2 | c |
| 3 | a |
A
B
Definition of Functions
A function is a relation in which each possible input value leads to exactly one output value.
It may be specified in 4 ways
By a Graph
By mapping
By displaying the pairs
By an equation
Input
Output
1
2
3
a
b
c
Input
Output
Input
Output
Output
Input
.
Ex. Determining If Menu Price Lists Are Functions
Item Price
Plain Donut ................................................ 1.49
Jelly Donut ................................................. 1.99
Chocolate Donut ........................................ 1.99
Is price a function of the item?
Is the item a function of the price?
Yes
No
Definition Functions
Ordered Pairs
Yes
Domain
Range
Yes
Domain
Range
No
Do the following relations define a function?
Find domain and range in the case the relation is a function.
No
Examples
Definition of Functions
Graph
A function is a relation in which each possible input value leads to exactly one output value.
Output
Input
Vertical Line Test
A graph represents a function if any vertical line drawn intersects the curve only once.
.
.
.
Domain
Range
The Domain is the input on the -axis
The Range is the output on the -axis
Letters used for functions include
Definition Functions
Graph
Domain
Range
Domain
Range
>
>
All real
numbers
All real
numbers
Not a Function
.
.
>
>
Is a Function
Is a Function
Examples
Not a Function
Definition of Functions
Equations
A function is a relation in which each possible input value leads to exactly one output value.
When you input one value of there is only one value for y
No
Yes
One input of yields
2 values of
Function Notation
The notation
defines a function named f.
is a function of
Input
Output
Instead of writing
we write
So is the name for the rule that defines
the output
Independent Variable
Dependent Variable
Independent Variable
Dependent Variable
.
Evaluating and Solving
Functions
Ordered pairs
Ex. If
Find: a.
b. Evaluate at 1.
Ex. Solve
For which is
From a table
Ex. Evaluate:
| 5 | 10 | 15 | 20 | 25 | 30 | |
| 10 | 9 | 8 | 7 | 6 | 5 |
Ex. Solve:
Evaluating and Solving
Functions
Evaluating and Solving
Functions
From a Graph
Independent
Variable
Dependent
Variable
.
Ex.
Evaluate: a.
b.
Evaluate: a.
Evaluating and Solving
Functions
From a Graph
>
>
Evaluate:
a.
b.
Solve:
a.
b.
Example
Evaluating and Solving
Functions
From an Equation
Find the value f(−2), where
Given the function ,
solve
+2
+2
A quadratic Equation
or
The value is -17
The Graph of a Function
is on the graph of
so is on the graph of
is always positive or 0
is all real numbers
is all positive numbers and 0
-intercept
-intercept
Understanding Graphs
>
>
For which values of is the function positive
Above the -axis
On
Example
For which values of is the function negative
Below the -axis
On
-intercept
-intercept
For which values of is the function above the line
On
Range
Domain
1) Find
2) Is positive or negative?
3) Find the -intercepts
4) Find the -intercept
5) Find where
6) Find where
7) How often does the line
intersect the graph of
8) For which values of is
Positive
3 times
and
Range
Domain
[-3.8, -3)
One to One Functions
One to One Functions
Mapping
Graph
Equation
A one-to-one function is a function in which each output value corresponds to exactly one input value. There are no repeated x- or y-values.
One to One Functions
Ordered Pairs
Domain
Range
Not a function
Yes
A one to one function
Domain
Range
A function that is not
one to one
Decide if the following relations define one to one functions?
Find the domain and range in every case you have a function.
Examples
One to One Functions
Mapping
1
2
3
Domain
Range
A
B
4
1
2
3
A
B
4
1
2
3
Domain
Range
A
B
4
1
2
3
Domain
Range
A
B
4
A one to one
function
Not a function
Not one to
One function
A one to one
function
Examples
One to One Functions
Graph
Horizontal Line Test
A graph represents a function if any horizontal line drawn intersects the curve only once.
.
.
A function that is not one to one
Not a function
A function that is
not one to one
A one-to-one function is a function in which each output value corresponds to exactly one input value. There are no repeated x- or y-values.
One to One Functions
Equation
A one-to-one function is a function in which each output value corresponds to exactly one input value. There are no repeated x- or y-values.
Is the area of a circle a function of its radius?
Solve for
The area is a function of radius r.
If yes, is the function one-to-one?
Area of a circle of radius r
In this problem, because radii are always positive. So there is exactly one solution.
The area of a circle is a one to one functions of the circle’s radius.
Ex. is a one to one function.
Library of Functions
Toolkit Functions
Local Behavior of Polynomial Functions
Learning Objectives
· Identify intercepts of polynomial functions in factored form
· Understand the relationship between degree, turning points, and x-intercepts
· Understand the intermediate value theorem
· Use factoring to find zeros of polynomial functions
· Identify zeros and their multiplicities from an equation or a graph
Identify intercepts of polynomial functions in factored form
1. Find the x- and y-intercepts of .
2. Find the x- and y-intercepts of .
3. Find the x- and y-intercepts of .
|
Interpreting Turning Points
A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).
A polynomial of degree n has n – 1 turning points. |
Understand the relationship between degree, turning points, and x-intercepts
4. Find the maximum number of turning points of the following functions:
a.
b.
c.
|
Intermediate Value Theorem
Let be a polynomial function. The Intermediate Value Theorem states that if and have opposite signs, then there exists at least one value between and for which . |
Understand the intermediate value theorem
5. Show that the function has a real zero between and
6. Show that the function has a real zero between and .
7. Show that the function has a real zero between and .
Use factoring to find zeros of polynomial functions
8. Find the x-intercepts of the following functions:
a.
b.
c.
Identify zeros and their multiplicities from an equation or a graph
9. Find all the zeros and their multiplicities for the function .
10. Find all the zeros and their multiplicities for the function .
11. Use the graph of the function of degree 4 below to find the zeros and their possible multiplicities.
ANSWER KEY
1.
2.
3.
4a. The function will have a maximum of 3 turning points.
4b. The function will have a maximum of 2 turning points.
4c. The function will have a maximum of 4 turning points.
5. . The sign change shows there is a zero between the given values.
6. . The sign change shows there is a zero between the given values.
7. . The sign change shows there is a zero between the given values.
8a.
8b.
8c.
9. have multiplicity of 1. has multiplicity 2.
10. has multiplicity 3, has multiplicity 2, has multiplicity 1.
11. . The graph touches the x-axis at both of the zeros so their multiplicity must be even and 2 since the degree of the function is 4.
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Applications of Quadratic Functions
Learning Objectives
· Find the domain and range of a quadratic function
· Determine the maximum and minimum values of quadratic functions
· Find the x- and y-intercepts of a quadratic function
· Use a quadratic function to model projectile motion
Find the domain and range of a quadratic function
1. Identify the domain and range of .
2. Identify the domain and range of
Determine the maximum and minimum values of quadratic functions
3. Determine if the following will have a minimum or a maximum value without solving.
a.
b.
4. Given , find the minimum or the maximum value and determine where it occurs.
5. Given , find the minimum or maximum value and determine where it occurs.
Find the x- and y-intercepts of a quadratic function
6. Find x- and y-intercepts of a quadratic function .
7. Find x- and y-intercepts of a quadratic function .
Use a quadratic function to model projectile motion
8. A ball is thrown upward and outward from a height of 6 feet. The height of the ball, , in feet, can be modeled by , where is the ball’s horizontal distance, in feet, from where it was thrown.
a. What is the maximum height of the ball and how far from where the ball is thrown does it occur?
b. How far does the ball travel horizontally before hitting the ground? Round to the nearest tenth of a foot.
ANSWER KEY
1. Domain: , Range:
2. Domain: , Range:
3a. Maximum
3b. Minimum
4. Maximum value is and it occurs at
5. Minimum value is and it occurs at
6. x-intercept: . y-intercept:
7. x-intercept: . y-intercept:
8a. Maximum height is 9.2 feet and it occurs 2 feet away from where the ball is thrown.
8b. 5.4
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Write and Graph Polynomial Functions
Learning Objectives
· Draw conclusions about a polynomial function from a graph
· Graph polynomial functions
· Write a formula for a polynomial function from a graph
· Determine equation of a polynomial given key information
Draw conclusions about a polynomial function from a graph
1. Given the graph below, what can be concluded about the polynomial function based on the intercepts and turning points?
2. Given the graph below, what can be concluded about the polynomial function based on the intercepts and turning points?
Graph polynomial functions
3. Graph .
4. Graph .
Write a formula for a polynomial function from a graph
5. Write a formula for the polynomial function given the graph below.
6. Write a formula for the polynomial function given the graph below.
Determine equation of a polynomial given key information
7. Find the equation of a polynomial of degree 4 with zeros at and , and
y-intercept .
8. Find the equation of a polynomial of degree 4 with zeros at and , and
y-intercept .
ANSWER KEY
1. The graph has 3 x-intercepts, which suggests that a function has degree 4 or greater. It also has 3 turning points, which suggests that a function has degree of 4 or greater. So it can be concluded that the function is even and has degree of 4 or greater.
2. The graph has 2 x-intercepts and 2 turning points. Also the two ends of the function are going in opposite directions. So the polynomial has a degree of 3 or greater and is an odd function.
3.
5.
6.
7.
8.
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