Obtaining Information from Graphs

You can obtain information about a function from its graph. At the right or left of a graph, you will find closed dots, open dots, or arrows.

• A

closed

dot indicates that the graph does not extend beyond this point and the point belongs to the graph.

• An

open

dot indicates that the graph does not extend beyond this point and the point does not belong to the graph.

• An

arrow

indicates that the graph extends indefinitely in the direction in which the arrow points.

Example: Obtaining Information from a Function’s Graph

Use the graph of the function

f

to answer the following questions.

a. What are the function values

f

(-1) and f (1)?

b. What is the domain of

f

(

x

)?

c. What is the range of

f

(

x

)?

5 4 3 2 1 Solution

of

f

a. Because (-1, 2) is a point on the graph , the y-coordinate, 2, is the value of the function at the x-coordinate, -1. Thus,

f

(-l) = 2.

-5 -4 -3 -2 -1 -1 -2 -3 -4 -5 1 2 3 4 5

Similarly, because (1, 4) is also a point on the graph of

f

, this indicates that

f

(1) = 4.

Example: Obtaining Information from a Function’s Graph

Solution

b. The open dot on the left shows that

x

= -3 is not in the domain of

f

. By contrast, the closed dot on the right shows that

x

= 6 is. We determine the domain of

f

by noticing that the points on the graph of

f

have

x

-coordinates between -3, excluding -3, and 6, including 6.

-5 -4 -3 -2 -1 -1 -2 -3 -4 -5 5 4 3 2 1 1 2 3 4 5

Thus, the domain of

f

is { x | -3 <

x

< 6} or the interval (-3, 6].

Example: Obtaining Information from a Function’s Graph

Solution

c. The points on the graph all have

y

-coordinates between -4, not including -4, and 4, including 4. The graph does not extend below

y

= -4 or above

y

= 4. Thus, the range of

f

is{

y

| -4 <

y

< 4} or the interval (-4, 4].

-5 -4 -3 -2 -1 -1 -2 -3 -4 -5 5 4 3 2 1 1 2 3 4 5

The Vertical Line Test for Functions

If any vertical line intersects a graph in more than one point, the graph does not define

y

as a function of

x

.

Example: Using the Vertical Line Test

Use the vertical line test to identify graphs in which

y

is a function of

x

.

a.

y

b.

y

c.

y

d.

y x x x

Solution

y

is a function of

x

for the graphs in (b) and (c). a.

y

b.

y

c.

y

d.

y x x y

is

not

a function since 2 values of

y

correspond to an

x

-value.

y

is a function of

x

.

x y

is a function of

x

.

x x y

is

not

a function since 2 values of

y

correspond to an

x

-value.

Increasing, Decreasing, and Constant Functions

A function is where

x

1 <

x

2

increasing

, then

f

on an interval if for any

x

1 , and

x

2 (

x

1 ) <

f

(

x

2 ).

A function is where

x

1 <

x

2

decreasing

, then

f

on an interval if for any

x

1 , and

x

2 (

x

1 ) >

f

(

x

2 ). A function is where

x

1 <

x

2

constant

, then

f

on an interval if for any

x

1 , and

x

2 (

x

1 ) =

f

(

x

2 ).

in the interval, in the interval, in the interval, (

x

2 ,

f

(

x

2 )) (

x

1 ,

f

(

x

1 )) Increasing

f

(

x

1 ) <

f

(

x

2 ) (

x

1 ,

f

(

x

1 )) (

x

2 ,

f

(

x

2 )) Decreasing

f

(

x

1 ) >

f

(

x

2 ) (

x

1 ,

f

(

x

1 )) (

x

2 ,

f

(

x

2 )) Constant

f

(

x

1 ) =

f

(

x

2 )

Example: Intervals on Which a Function Increases, Decreases, or Is Constant Describe the increasing, decreasing, or constant behavior of each function whose graph is shown.

a.

-5 -4 -3 -2 -1 -1 -2 -3 -4 -5 5 4 3 2 1 1 2 3 4 5

b.

1 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 5 4 3 1 2 3 4 5 Solution

a. Take note as to when the function changes direction. The function is decreasing on the interval (-oo, 0), increasing on the interval (0, 2), and decreasing on the interval (2, oo).

Example: Intervals on Which a Function Increases, Decreases, or Is Constant Describe the increasing, decreasing, or constant behavior of each function whose graph is shown.

a.

5 4 3 2 1

b.

5 4 3 1 1 5 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 1 2 3 4 5 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 2 3 4 Solution

b. The graph indicates that the function is defined in two pieces. The part of the graph to the left of the

y

-axis shows that the function is constant with an open endpoint on the right. So the function is constant on the interval ( oo, 0). The part to the right of the

y

-axis shows that the function is increasing on the interval with a closed dot on the left. So the function is increasing on the interval [0, oo).

Definition of Even and Odd Functions

The function f is an

even function

if

f

(-

x

) =

f

(

x

) for all

x

in the domain of

f

.

The right side of the equation of an even function does not change if

x

is replaced with -

x

. The function

f

is an

odd function

if

f

(-

x

) = -

f

(

x

) for all

x

in the domain of

f

.

Every term in the right side of the equation of an odd function changes sign if

x

is replaced by -

x

.

Example

Identify the following function as even, odd, or neither: f(x) = 3x 2 - 2.

Solution : We use the given function’s equation to find f(-x).

f(-x) = 3(-x) 2 -2 = 3x 2 - 2.

The right side of the equation of the given function did not change when we replaced x with -x. Because f(-x) = f(x), f is an even function.

Example

Identify the following function as even, odd, or neither: g(x) = x 2 + x - 2.

Solution :

We use the given function’s equation to find g(-x).

g(-x) = (-x) 2 + (-x) - 2 = x 2 - x - 2.

The right side of the equation of the given function changed when we replaced x with -x. Because g( x) ≠ g(x), f is not an even function.

Next we check to see if the function is odd where -g(x) = g(-x).

-g(x) = -(x 2 + x - 2) = -x 2 x + 2 ≠ x 2 - x - 2 = g(-x) So the function is not an odd fiunction. Therefore the function is neither.

Even Functions and y-Axes Symmetry

The graph of an even function in which

f

(-

x

) =

f

symmetric with respect to the

y

-axis.

(

x

) is

Odd Functions and Origin Symmetry

The graph of an odd function in which

f

(-

x

) = symmetric with respect to the origin.

f

(

x

) is

Definitions of Relative Maximum and Relative Minimum

1. A function value

f(a)

is a

relative maximum

of

f

if there exists an open interval about that

f(a)

>

f(x)

for all

x a

such in the open interval.

2. A function value

f(b)

is a

relative minimum

if there exists an open interval about

b

such of

f

that

f(b)

<

f(x)

for all

x

in the open interval.

The Average Rate of Change of a Function

• Let

(x 1 , f(x 1 )

) and (

x 2 , f(x 2 ))

the graph of a function

f

.

be distinct points on • The

average rate of change of f

from x

1

to x

2

is

f

(

x

2 ) 

x

2 

f x

1 (

x

1 )