2.4 graphs of functions

2.4 Graphs of Functions

The graph of a function is the graph of its ordered pairs.

Graphing a Function by Plotting Points

feature in your

graphing utility. You can use it to create

tables of coordinates for a function. You

will need to enter the equation of the

function and sepcify the starting value for

x

Use the TABLE

and the increment betweenTblStart

successive -valuesx Tbl .

Steps for Graphing an Equation Using a Graphing Utility

: Solve the equation for in

terms

St

o

ep

.

1

f

y

x

Steps for Graphing an Equation Using a Graphing Utility

: Get in to the graphing mode of your

graphing utility. The screen will usually

display , prompting you to

enter the expression involving that you

St

found i

ep 2

y

x

n step 1.

Steps for Graphing an Equation Using a Graphing Utility

: Select the viewing window.

Adjust the viewing window until

a complete graph is obta

Step 3

ined.

Ex 1: Graphing a Function by Plotting Points

Graph the function: ( ) 2f x x

Hint: To do so, use integer values of x from

the set {0,1,4,9,16,25} to obtain six ordered

pairs. Plot each ordered pair and draw a

smooth curve through the points. Use the graph

to specify the function's domain and range.

SolutionSet up a table of coordinates.

x

0

1

4

9

16

25

( ) 2f x x ( , ) or ( , ( )x y x f x

(0) 0 2 2f (0, 2)

(1) 1 2 3f

(4) 4 2 4f

(9) 9 2 5f

(16) 16 2 6f

(25) 25 2 7f

(1,3)

(4,4)

(9,5)

(16,6)

(25,7)

Solution Contd.

Now we plot the six points and draw a smoth

curve through them. The points on the graph

of have -coordinates that start at zero and

extend indefinitely far to the right. Thus, the

domain consists of al

f x

l real numbers greater

than or equal to 0, represented by [0, ).

Solution Contd.

Graph of ( ) 2f x x

Solution Contd.

Similarly, the points on the graph have

-coordinates that start at 2 and extend

indefinitely upward. Thus, the range

consists of all real numbers greater

than or equal to 2, represented by [2, ).

y

Solution Contd.Graph of ( ) 2f x x

Practice Exercise

Graph the function ( ) | | 1. Use the

integer values 3, 2, 1,0,1,2,3 to

obtain ordered pairs. Use the graph to

specify the function's domain and range.

f x x

x

AnswerGraph of ( ) | | 1f x x

Domain:

Range:

( , )

[1, )

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.

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

Obtaining Information From Graphs 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.

Ex 2: Obtaining Information From a Function’s Graph

Use the graph of the function to answer

the following questions.

a. What are the function values ( 1),

(0) and (1)?

b. What is the domain of ?

c. What is the range of ?

f

f

f f

f

f

Solution of part a.

Because ( 1,4), (0,3), and (1,0)

are points on the graph of ,

the -coordinate, 4 is the ( 1),

the -coordinate, 3 is the (0), and

the -coordinate, 0 is the (1).

f

y f

y f

y f

Thus, ( 1) 4, (0) 3, (1) 0.f f f

Solution of part b.The arrow on the left shows that the

graph extends indefinitely to left.

The close dot on the right shows that

2 is in the domain of and does

not extend beyond this point. Thus, the

domain of is {

x f f

f x

| 2} or the interval

( , 2].

x

Solution of part c.

The points on the graph all have

-coordinates from to 4,

including 4. The graph does not

extends above 4.

Thus, the range of is { | 4}

or the interval ( , 4].

y

y

f y x

The Vertical Line Test

If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.

Ex 3: Using the Vertical Line Test

Use the vertical line test to identify graphs

in which is a function of .y x

a. b.

Solution

is a function of .y x

a.b.

is not a function of .

Two values of

correspond to an -value.

y x

y

x

Increasing Function

A function isincreasing on aninterval if for any x1

and x2 in the interval,

where x1<x2, then

f(x1)<f(x2).

1 1( , ( ))x f x

2 2( , ( ))x f x

1x 2x

Decreasing Function

A function isDecreasing on aninterval if for any x1

and x2 in the interval,

where x1<x2, then

f(x1)>f(x2).

1 1( , ( ))x f x

2 2( , ( ))x f x

1x 2x

Constant Function

A function isconstant on aninterval if for any x1

and x2 in the interval,

where x1<x2, then

f(x1)=f(x2).

1 1( , ( ))x f x

2 2( , ( ))x f x

1x 2x

Ex 4: Intervals on Which a Function Increases, Decreases, or Is Constant

Describe the increasing, decreasing, or

constant behavior of the function whose

graph is shown.

SolutionAlthough the function's equations are not

given, the graph indicates that the function

is defined in four pieces.

The function is constant on the intervals

( , 4) and (2, ).

The function is decreasing on

the interval

( 4,2).

Even Function

The function is an if

for all in the domain of .

The right side of the equation of an even

function does not change

even functi

if is replaced

with

(

.

on

) ( )

f

xf x

x

f

x

x f

Odd Function

The function is an if

for all in the domain of .

Every term in the right side of the equation

of an odd function changes sign

odd functio

if is

re

n

( ) (

placed .

)

with

f

x f

x

x x

x

f f

Identifying Even or Odd Functions

3

2

2 4

Identify each of the following functions

as even, odd, or neither.

a. ( )

b. ( )

c. ( ) 2

f x x x

g x x x

h x x x

Solution

In each case, replace with and simplify.

If the right side of the equation stays the same,

the function is even. If every term on the right

changes sign, the function is odd.

x x

Solution for part a.

3 ( ) Given func ion tx x xf

3( )f x x x

Replace with .x x 3( )x x xf

3( )f x x x

( ) ( )f x f x Factor

Simplify.

3Replace with ( ).x x f x

Thus, is an odd function.f

Solution for part b.

2 ( ) Given func ion tx x xg

2( )g x x x

Replace with .x x 2( )x x xg

Thus, ( ) ( ) or ( ) ( ).g x g x g x g x Simplify.

We conclude that is neither an even

nor an odd function.

g

Solution for part c.2 4 ( ) 2 Given f unctio nx x xh

2 4( ) 2h x x x

Replace with .x x 2 4( 2)x xh x

( ) ( )h x h x

Simplify.

2 4Replace 2 with h( ).x x x

Thus, is an even function.h

Even Functions and y-Axis Symmetry

The graph of an even function in which

( ) ( ) is symmetric with respect

to the -axis.

f x f x

y

Odd Functions and Origin Symmetry

The graph of an odd function in which

( ) ( ) is symmetric with respect

to the origin.

f x f x

Graphs of Common Functions

Use a graphing utility to verify the following six graphs.

Constant Function Domain: Range: the single

number Constant on

Even function

( )f x c

(,)

c

(,)

Identity Function( )f x x

Domain:Range:Increasing on

Odd function

( , )

( , )

( , )

Standard Quadratic Function

Domain:

Range:

Decreasing on and

increasing on

Even function

( , )

[0, )( ,0) (0, )

2( )f x x

Standard Cubic Function

3( )f x x

Domain:

Range:

Increasing on

Odd function

( , )

( , )

( , )

Square Root Function

( )f x x

Domain:

Range:

Increasing on

Neither even nor odd

[0, )[0, )

[0, )

Absolute Value Function

( ) | |f x x

Domain:Range:

Decreasing on and

incerasing on

Even function

( , ) ( , )

( ,0) (0, )

Greatest Integer Function

( ) int( )

Greatest integer less than or equal to

f x x x

x

The graph of the

greatest integer

function ( ) int( ).f x x