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Objective To be able to identify a function and to be able to graphically represent functions. Purpose To help describe input-output relations in real-world applications and to use functions to model and solve real-life problems.

1.
Functions and their Graphs

Lesson 2

Sections 1.2 and 1.3

Lesson 2

Sections 1.2 and 1.3

2.
Directions for Open–Minded

Questions – Warm Up

Everyone silently read through both problems.

Write down what you know.

Read the two questions again.

Group Discuss

You have 5 minutes to read and discuss and 10 minutes or less to show solutions.

Questions – Warm Up

Everyone silently read through both problems.

Write down what you know.

Read the two questions again.

Group Discuss

You have 5 minutes to read and discuss and 10 minutes or less to show solutions.

3.
Solve the following. Try to find

multiple solution paths!

In an all-adult apartment building, 2/3 of the men are

married to 3/5 of the women. What fraction of the

residents is married?

A farmer had hens and rabbits. These animals have

50 heads and 140 feet. How many hens and rabbits

does the farmer have?

multiple solution paths!

In an all-adult apartment building, 2/3 of the men are

married to 3/5 of the women. What fraction of the

residents is married?

A farmer had hens and rabbits. These animals have

50 heads and 140 feet. How many hens and rabbits

does the farmer have?

4.
In an all-adult apartment building, 2/3 of the men are married

to 3/5 of the women. What fraction of the residents is married?

to 3/5 of the women. What fraction of the residents is married?

5.
A farmer had hens and rabbits. These animals have 50 heads

and 140 feet. How many hens and rabbits does the farmer have?

and 140 feet. How many hens and rabbits does the farmer have?

6.
Objective

To be able to identify a function and to be

able to graphically represent functions.

Purpose

To help describe input-output relations in

real-world applications and to use functions

to model and solve real-life problems.

To be able to identify a function and to be

able to graphically represent functions.

Purpose

To help describe input-output relations in

real-world applications and to use functions

to model and solve real-life problems.

7.
Relation – pairs of quantities that are related

to each other

Example: The area A of a circle is related to

its radius r by the formula

2

A r .

to each other

Example: The area A of a circle is related to

its radius r by the formula

2

A r .

8.
There are different kinds of relations.

When a relation matches each item from one

set with exactly one item from a different set

the relation is called a function.

When a relation matches each item from one

set with exactly one item from a different set

the relation is called a function.

9.
Definition of a Function

A function is a relationship between two

variables such that each value of the first

variable is paired with exactly one value of

the second variable.

The domain is the set of permitted x values.

The range is the set of found values of y.

These can be called images.

A function is a relationship between two

variables such that each value of the first

variable is paired with exactly one value of

the second variable.

The domain is the set of permitted x values.

The range is the set of found values of y.

These can be called images.

10.
Is it a Function?

For each x, there is

only one value of y. Domain, x Range, y

1 -3.6

Therefore, it IS a 2 -3.6

function. 3 4.2

4 4.2

5 10.7

6 12.1

52 52

For each x, there is

only one value of y. Domain, x Range, y

1 -3.6

Therefore, it IS a 2 -3.6

function. 3 4.2

4 4.2

5 10.7

6 12.1

52 52

11.
Is it a function?

Three different y-

values (7, 8, and 10) Domain, x Range, y

are paired with one x- 3 7

value. 3 8

3 10

Therefore, it is NOT a

4 42

function

10 34

11 18

52 52

Three different y-

values (7, 8, and 10) Domain, x Range, y

are paired with one x- 3 7

value. 3 8

3 10

Therefore, it is NOT a

4 42

function

10 34

11 18

52 52

12.
Is it a function? State the domain and range.

{(5, 8), (6, 7), (3, -1), (4, 2), (5, 9), (12, -2)

No. The x-value of 5 is paired with two

different y-values.

Domain: (5, 6, 3, 4, 12)

Range: (8, 7, -1, 2, 9, -2)

{(5, 8), (6, 7), (3, -1), (4, 2), (5, 9), (12, -2)

No. The x-value of 5 is paired with two

different y-values.

Domain: (5, 6, 3, 4, 12)

Range: (8, 7, -1, 2, 9, -2)

13.
Vertical Line Test

Used to determine if a graph is a function.

If a vertical line intersects the graph at more than

one point, then the graph is NOT a function.

NOT a Function

Used to determine if a graph is a function.

If a vertical line intersects the graph at more than

one point, then the graph is NOT a function.

NOT a Function

14.
Is it a function? Give the domain

and range.

FUNCTION

Domain : 4,2

Range : 4,4

and range.

FUNCTION

Domain : 4,2

Range : 4,4

15.
Give the Domain and Range.

Domain : x 1 Domain : 2 x 2

Range : y 2 Range : 0 y 3

Domain : x 1 Domain : 2 x 2

Range : y 2 Range : 0 y 3

16.
Functional Notation

We have seen an equation written in the form

y = some expression in x.

Another way of writing this is to use

functional notation.

For Example, you could write y = x²

as f(x) = x².

We have seen an equation written in the form

y = some expression in x.

Another way of writing this is to use

functional notation.

For Example, you could write y = x²

as f(x) = x².

17.
Functional Notation: Find the

f ( x ) 3 x x 2

2

f ( x) x x 22

f ( 3) f ( m 3)

2

2

3 3 3 2 m 3 m 3 2

27 3 2 m 3 m 3 m 3 2

30 2

m 2 3m 3m 9 m 3 2

m 2 5m 8

f ( x ) 3 x x 2

2

f ( x) x x 22

f ( 3) f ( m 3)

2

2

3 3 3 2 m 3 m 3 2

27 3 2 m 3 m 3 m 3 2

30 2

m 2 3m 3m 9 m 3 2

m 2 5m 8

18.
Let’s look at Functions

19.
Find: f ( 2) g ( 4)

f ( x) g ( x)

f ( x) g ( x)

20.
Find: f ( 5) g ( 0 )

f ( x) g ( x)

f ( x) g ( x)

21.
Find: f ( 4) g ( 1)

f ( x) g ( x)

f ( x) g ( x)

22.
Find: f ( 2) g ( 0)

f ( x) g ( x)

f ( x) g ( x)

23.

24.
A piecewise-defined function is a function that is

defined by two or more equations over a specified

domain.

The absolute value function f x x

can be written as a piecewise-defined function.

The basic characteristics of the absolute value

function are summarized on the next page.

defined by two or more equations over a specified

domain.

The absolute value function f x x

can be written as a piecewise-defined function.

The basic characteristics of the absolute value

function are summarized on the next page.

25.
Absolute Value Function is a

Piecewise Function

Piecewise Function

26.
Evaluate the function when x = -1 and 0.

27.
Domain of a Function

28.
The domain of a function can be implied by

the expression used to define the function

The implied domain is the set of all real

numbers for which the expression is defined.

For example,

the expression used to define the function

The implied domain is the set of all real

numbers for which the expression is defined.

For example,

29.
The function has an implied

domain that consists of all real x other than

x = ±2

The domain excludes x-values that result

in division by zero.

domain that consists of all real x other than

x = ±2

The domain excludes x-values that result

in division by zero.

30.
Another common type of implied domain is

that used to avoid even roots of negative

numbers.

EX:

is defined only for x 0.

The domain excludes x-values that result

in even roots of negative numbers.

that used to avoid even roots of negative

numbers.

EX:

is defined only for x 0.

The domain excludes x-values that result

in even roots of negative numbers.

31.
Graphs of Functions

Lesson 3

Lesson 3

32.
Objective:

To graph a function using domain and

range, even or odd, relative min/max.

Purpose:

To introduce methods to help graph a

function.

To graph a function using domain and

range, even or odd, relative min/max.

Purpose:

To introduce methods to help graph a

function.

33.
Domain & Range of a Function

What is the

domain of

the graph of

the function

f?

A : 1,4

What is the

domain of

the graph of

the function

f?

A : 1,4

34.
Domain & Range of a Function

What is the

range of

the graph of

the function

f?

5,4

What is the

range of

the graph of

the function

f?

5,4

35.
Domain & Range of a Function

Find f 1 and f 2 .

f 1 5

f 2 4

Find f 1 and f 2 .

f 1 5

f 2 4

36.
Let’s look at domain and range of a

function using an algebraic approach.

Then, let’s check it with a graphical

approach.

function using an algebraic approach.

Then, let’s check it with a graphical

approach.

37.
Find the domain and range of

f x x 4.

Algebraic Approach

The expression under the radical can not be negative.

Therefore, x 4 0. Domain

A : x 4 Since the domain is never negative the

or range is the set of all nonnegative real

4, numbers.

A : y 0

or Range

0,

f x x 4.

Algebraic Approach

The expression under the radical can not be negative.

Therefore, x 4 0. Domain

A : x 4 Since the domain is never negative the

or range is the set of all nonnegative real

4, numbers.

A : y 0

or Range

0,

38.
Find the domain and range of

f x x 4.

Graphical Approach

f x x 4.

Graphical Approach

39.
Increasing and Decreasing

40.
The more you know about the graph of

a function, the more you know about

the function itself.

Consider the graph on the next slide.

a function, the more you know about

the function itself.

Consider the graph on the next slide.

41.
Falls from x = -2 to x = 0.

Is constant from x = 0 to

x = 2.

Rises from x = 2 to x = 4.

Is constant from x = 0 to

x = 2.

Rises from x = 2 to x = 4.

42.
Ex: Find the open intervals on which the

function is increasing, decreasing, or constant.

Increases over

the entire real

line.

function is increasing, decreasing, or constant.

Increases over

the entire real

line.

43.
Ex: Find the open intervals on which the

function is increasing, decreasing, or constant.

INCREASING :

, 1 and 1,

DECREASING :

1,1

function is increasing, decreasing, or constant.

INCREASING :

, 1 and 1,

DECREASING :

1,1

44.
Ex: Find the open intervals on which the

function is increasing, decreasing, or constant.

INCREASING :

,0

CONSTANT :

0,2

DECREASING :

2,

function is increasing, decreasing, or constant.

INCREASING :

,0

CONSTANT :

0,2

DECREASING :

2,

45.
Relative Minimum and

Maximum Values

Maximum Values

46.
Relative Min/Max

The point at which a function changes

its increasing, decreasing, or constant

behavior are helpful in determining the

relative maximum or relative

minimum values of a function.

The point at which a function changes

its increasing, decreasing, or constant

behavior are helpful in determining the

relative maximum or relative

minimum values of a function.

47.
General Points – We’ll find

EXACT points later……

EXACT points later……

48.
Approximating a Relative

Example: Use a GDC to approximate

the relative minimum of the function

given by

2

f x 3 x 4 x 2.

Example: Use a GDC to approximate

the relative minimum of the function

given by

2

f x 3 x 4 x 2.

49.
2

f x 3 x 4 x 2 .

Put the function into the “y = “ the

press zoom 6 to look at the graph.

Press 2nd Calc, 3:minimum, left bound,

right bound, enter at the lowest point.

f x 3 x 4 x 2 .

Put the function into the “y = “ the

press zoom 6 to look at the graph.

Press 2nd Calc, 3:minimum, left bound,

right bound, enter at the lowest point.

50.
Use a GDC to approximate the relative

minimum and relative maximum of the

function given by

3

f x x x.

minimum and relative maximum of the

function given by

3

f x x x.

51.
Relative Minimum

(-0.58, -0.38)

(-0.58, -0.38)

52.
Relative Maximum

(0.58, 0.38)

(0.58, 0.38)

53.
Step Functions and

Piecewise-Defined Functions

Piecewise-Defined Functions

54.
Because of the vertical jumps, the greatest integer function is an example

of a step function.

of a step function.

55.
Let’s graph a Piecewise-

Defined Function

Sketch the graph of

2 x 3, x 1

f x

x 4, x 1

Notice when open

dots and closed

dots are used. Why?

Defined Function

Sketch the graph of

2 x 3, x 1

f x

x 4, x 1

Notice when open

dots and closed

dots are used. Why?

56.
Even and Odd Functions

57.

58.
Algebraically

Let’s look at the graphs again and see if this applies.

Let’s look at the graphs again and see if this applies.

59.
☺ ☺

60.
Determine whether each function is

even, odd, or neither.

even, odd, or neither.

61.
Graphical –

Algebraic Symmetric to

Origin

Algebraic Symmetric to

Origin

62.
Graphical –

Symmetric to y-

axis

Symmetric to y-

axis

63.
Graphical – NOT

Algebraic Symmetric to

origin OR y-axis.

Algebraic Symmetric to

origin OR y-axis.

64.
You Try

Is the function

f x x

Even, Odd, of Neither?

Is the function

f x x

Even, Odd, of Neither?

65.
f x x

Symmetric about the y-axis.

Symmetric about the y-axis.