1.2 Basics of Functions and Their Graphs Notes

Basics of Functions and Their Graphs

Section 1.2 Basics of Functions and Their Graphs 147

1.2 Basics of Functions and Their GraphsObjectives

! Find the domain and rangeof a relation.

" Determine whether a relationis a function.

# Determine whether anequation represents afunction.

$ Evaluate a function.

% Graph functions by plottingpoints.

& Use the vertical line testto identify functions.

' Obtain information abouta function from its graph.

( Identify the domain and rangeof a function from its graph.

) Identify intercepts froma function’s graph.

Sec t i on

! Find the domain and range of a relation.

Magnified 6000 times, this color-scanned imageshows a T-lymphocyte blood cell (green) infectedwith the HIV virus (red). Depletion of the num-

ber of T cells causes destruction of the immunesystem.

Definition of a Relation

A relation is any set of ordered pairs. The set of all first components of theordered pairs is called the domain of the relation and the set of all secondcomponents is called the range of the relation.

Finding the Domain and Range of a Relation

Find the domain and range of the relation:

Solution The domain is the set of all first components. Thus, the domain is5Smith, Johnson, Williams, Brown, Jones6.1Brown, 0.621%2, 1Jones, 0.621%26.51Smith, 1.006%2, 1Johnson, 0.810%2, 1Williams, 0.699%2,

EXAMPLE 1

The average number of T cellsin a person with HIV is a

function of time after infection.In this section, you will beintroduced to the basics offunctions and their graphs. Wewill analyze the graph of a

function using an example thatillustrates the progression of HIV

and T cell count. Much of our work inthis course will be devoted to the

important topic of functions and how theymodel your world.

RelationsThe top five U.S. last names shown in Table 1.1 account for nearly 4% of the entirepopulation. The table indicates a correspondence between a last name and the percentage of Americans who share that name. We can write this correspondenceusing a set of ordered pairs:

The mathematical term for a set of ordered pairs is a relation.

{(Smith, 1.006%), (Johnson, 0.810%), (Williams, 0.699%),(Brown, 0.621%), (Jones, 0.621%)}.

These braces indicate we are representing a set.

Table 1.1 Top U.S. LastNames

Name % of All Names

Smith 1.006%

Johnson 0.810%

Williams 0.699%

Brown 0.621%

Jones 0.621%

Source: Russell Ash, The Top 10 of Everything

P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 147


Determining Whether an Equation Represents a Function

Determine whether each equation defines as a function of

a. b.

Solution Solve each equation for in terms of If two or more values of canbe obtained for a given the equation is not a function.

a. This is the given equation.

Solve for by subtracting from both sides.

Simplify.

From this last equation we can see that for each value of there is one andonly one value of For example, if then The equationdefines as a function of

b. This is the given equation.

Isolate by subtracting from both sides.

Simplify.

Apply the square root property: If then

The in this last equation shows that for certain values of (all valuesbetween and 2), there are two values of For example, if then

For this reason, the equation does not define as afunction of

Check Point 3 Solve each equation for and then determine whether theequation defines as a function of

a. b.

Function NotationIf an equation in and gives one and only one value of for each value of then the variable is a function of the variable When an equation represents afunction, the function is often named by a letter such as or Anyletter can be used to name a function. Suppose that names a function. Think ofthe domain as the set of the function’s inputs and the range as the set of thefunction’s outputs. As shown in Figure 1.16, input is represented by and theoutput by The special notation read “ of ” or “ at ” representsthe value of the function at the number

Let’s make this clearer by considering a specific example. We know that theequation

defines as a function of We’ll name the function Now, we can apply our newfunction notation.

Input Output Equation

We read this equationas “f of x equals

0.013x 2 0.21x � 8.7.”

x f(x) f(x)=0.013x2-0.21x+8.7

f.x.y

y = 0.013x2 - 0.21x + 8.7

x.x,fxff(x),f1x2. x

fH.f, g, h, F, G,

x.yx,yyx

x2 + y2 = 1.2x + y = 6

x:yy

x.yy = ; 34 - 12 = ; 23.

x = 1,y.-2x;

u = ; 1d .u2 = d,y = ; 34 - x2

y2 = 4 - x2

x2y2x2 + y2 - x2 = 4 - x2

x2 + y2 = 4

x.yy = 4 - 12 = 3.x = 1,y.

x,

y = 4 - x2

x2yx2 + y - x2 = 4 - x2

x2 + y = 4

x,yx.y

x2 + y2 = 4.x2 + y = 4

x:y

EXAMPLE 3

! Evaluate a function.

Input x

Outputf(x )

f

Figure 1.16 A “functionmachine” with inputs and outputs

Study TipThe notation does not mean “ times ” The notation describesthe value of the function at x.

x.ff1x2

P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 151

Use the first fiveordered pairs

(x, y 1) to check thefirst table.


(x, y 2) to check thesecond table.

We entered 2for the startingx-value and 1 as

an incrementbetween x-values

to check our tablesin Example 5.

Checking Tables

We selected this viewing rectangle,or window, to match Figure 1.18.

[ 6, 6, 1] by [ 6, 9, 1]

y 2 � 2x � 4

y 1 � 2x

Checking Graphs

TechnologyWe can use a graphing utility to check the tables and the graphs in Example 5 for the functions

Enter y 1 = 2xin the y � screen.

f(x)=2x and g(x)=2x+4.

Enter y 2 = 2x � 4in the y � screen.

Check Point 5 Graph the functions and in the samerectangular coordinate system. Select integers for starting with and endingwith 2. How is the graph of related to the graph of

The Vertical Line TestNot every graph in the rectangular coordinate system is the graph of a function.Thedefinition of a function specifies that no value of can be paired with two or moredifferent values of Consequently, if a graph contains two or more different pointswith the same first coordinate, the graph cannot represent a function. This isillustrated in Figure 1.19. Observe that points sharing a common first coordinate arevertically above or below each other.

This observation is the basis of a useful test for determining whether a graphdefines as a function of The test is called the vertical line test.x.y

y.x

f?g-2x,

g1x2 = 2x - 3f1x2 = 2x

1

12345

2 3 4 5

1 2 3 4 5 1 2 3 4 5

y

x

(0, 1)

(0, 1)

(0, 0)

Figure 1.19 is not a functionof because 0 is paired with threevalues of namely, 1, 0, and -1.y,

xy

a. b. c. d.

x

y

x

y

x

y

x

y

The Vertical Line Test for Functions

If any vertical line intersects a graph in more than one point, the graph does notdefine as a function of x.y

Using the Vertical Line Test

Use the vertical line test to identify graphs in which is a function of x.y

EXAMPLE 6

! Use the vertical line test to identify functions.

154 Chapter 1 Functions and Graphs

P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 154


(x, y 1) to check thefirst table.


(x, y 2) to check thesecond table.

We entered 2for the startingx-value and 1 as

an incrementbetween x-values

to check our tablesin Example 5.

Checking Tables

We selected this viewing rectangle,or window, to match Figure 1.18.

[ 6, 6, 1] by [ 6, 9, 1]

y 2 � 2x � 4

y 1 � 2x

Checking Graphs

TechnologyWe can use a graphing utility to check the tables and the graphs in Example 5 for the functions

Enter y 1 = 2xin the y � screen.

f(x)=2x and g(x)=2x+4.

Enter y 2 = 2x � 4in the y � screen.

Check Point 5 Graph the functions and in the samerectangular coordinate system. Select integers for starting with and endingwith 2. How is the graph of related to the graph of

The Vertical Line TestNot every graph in the rectangular coordinate system is the graph of a function.Thedefinition of a function specifies that no value of can be paired with two or moredifferent values of Consequently, if a graph contains two or more different pointswith the same first coordinate, the graph cannot represent a function. This isillustrated in Figure 1.19. Observe that points sharing a common first coordinate arevertically above or below each other.

This observation is the basis of a useful test for determining whether a graphdefines as a function of The test is called the vertical line test.x.y

y.x

f?g-2x,

g1x2 = 2x - 3f1x2 = 2x

1

12345

2 3 4 5

1 2 3 4 5 1 2 3 4 5

y

x

(0, 1)

(0, 1)

(0, 0)

Figure 1.19 is not a functionof because 0 is paired with threevalues of namely, 1, 0, and -1.y,

xy

a. b. c. d.

x

y

x

y

x

y

x

y

The Vertical Line Test for Functions

If any vertical line intersects a graph in more than one point, the graph does notdefine as a function of x.y

Using the Vertical Line Test

Use the vertical line test to identify graphs in which is a function of x.y

EXAMPLE 6

! Use the vertical line test to identify functions.


P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 154

1.2


1.2 Basics of Functions and Their GraphsObjectives

! Find the domain and rangeof a relation.

" Determine whether a relationis a function.

# Determine whether anequation represents afunction.

$ Evaluate a function.

% Graph functions by plottingpoints.

& Use the vertical line testto identify functions.

' Obtain information abouta function from its graph.

( Identify the domain and rangeof a function from its graph.

) Identify intercepts froma function’s graph.

Sec t i on

! Find the domain and range of a relation.

Magnified 6000 times, this color-scanned imageshows a T-lymphocyte blood cell (green) infectedwith the HIV virus (red). Depletion of the num-

ber of T cells causes destruction of the immunesystem.

Definition of a Relation

A relation is any set of ordered pairs. The set of all first components of theordered pairs is called the domain of the relation and the set of all secondcomponents is called the range of the relation.

Finding the Domain and Range of a Relation

Find the domain and range of the relation:

Solution The domain is the set of all first components. Thus, the domain is5Smith, Johnson, Williams, Brown, Jones6.1Brown, 0.621%2, 1Jones, 0.621%26.51Smith, 1.006%2, 1Johnson, 0.810%2, 1Williams, 0.699%2,

EXAMPLE 1

The average number of T cellsin a person with HIV is a

function of time after infection.In this section, you will beintroduced to the basics offunctions and their graphs. Wewill analyze the graph of a

function using an example thatillustrates the progression of HIV

and T cell count. Much of our work inthis course will be devoted to the

important topic of functions and how theymodel your world.

RelationsThe top five U.S. last names shown in Table 1.1 account for nearly 4% of the entirepopulation. The table indicates a correspondence between a last name and the percentage of Americans who share that name. We can write this correspondenceusing a set of ordered pairs:

The mathematical term for a set of ordered pairs is a relation.

{(Smith, 1.006%), (Johnson, 0.810%), (Williams, 0.699%),(Brown, 0.621%), (Jones, 0.621%)}.

These braces indicate we are representing a set.

Table 1.1 Top U.S. LastNames

Name % of All Names

Smith 1.006%

Johnson 0.810%

Williams 0.699%

Brown 0.621%

Jones 0.621%

Source: Russell Ash, The Top 10 of Everything

P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 147


SmithJohnsonWilliamsBrownJones

1.006%0.810%0.699%0.621%

Domain Range

Figure 1.14(b) Percents correspondto names.

SmithJohnsonWilliamsBrownJones

1.006%0.810%0.699%0.621%

Domain Range

Figure 1.14(a) Names correspond topercents.

FunctionsTable 1.1, repeated in the margin, shows the top five U.S. last names and the percentageof Americans who share those names. We’ve used this information to define tworelations. Figure 1.14(a) shows a correspondence between last names and percentssharing those names. Figure 1.14(b) shows a correspondence between percents sharinglast names and those last names.Table 1.1 Top U.S. Last

Names

Name % of All Names

Smith 1.006%

Johnson 0.810%

Williams 0.699%

Brown 0.621%

Jones 0.621%

A relation in which each member of the domain corresponds to exactly onemember of the range is a function. Can you see that the relation in Figure 1.14(a) isa function? Each last name in the domain corresponds to exactly one percent in therange. If we know the last name, we can be sure of the percentage of Americanssharing that name. Notice that more than one element in the domain cancorrespond to the same element in the range: Brown and Jones are both shared by0.621% of Americans.

Is the relation in Figure 1.14(b) a function? Does each member of the domaincorrespond to precisely one member of the range? This relation is not a functionbecause there is a member of the domain that corresponds to two different membersof the range:

(0.621%, Brown) (0.621%, Jones).

The member of the domain 0.621% corresponds to both Brown and Jones in the range.If we know that the percentage of Americans sharing a last name is 0.621%, we cannotbe sure of that last name. Because a function is a relation in which no two ordered pairshave the same first component and different second components, the ordered pairs(0.621%, Brown) and (0.621%, Jones) are not ordered pairs of a function.

(0.621%, Brown) (0.621%, Jones)

Same first component

Different second components

Definition of a FunctionA function is a correspondence from a first set, called the domain, to a secondset, called the range, such that each element in the domain corresponds to exactlyone element in the range.

In Check Point 1, we considered a relation that gave a correspondencebetween years after 1970 and the percentage of first-year college women claimingno religious affiliation. Can you see that this relation is a function?

However, Example 2 illustrates that not every correspondence between sets is afunction.

{(0, 9.1), (10, 6.7), (20, 10.7), (30, 13.2), (36, 17.4)}

corresponds to exactly one element in the range.

Each element in the domain

! Determine whether a relationis a function.

P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 149

Determining Whether a Relation Is a Function

Determine whether each relation is a function:

a. b.

Solution We begin by making a figure for each relation that shows the domainand the range (Figure 1.15).

a. Figure 1.15(a) shows that every element in the domain corresponds to exact-ly one element in the range. The element 1 in the domain corresponds to theelement 6 in the range. Furthermore, 2 corresponds to 6, 3 corresponds to 8,and 4 corresponds to 9. No two ordered pairs in the given relation have thesame first component and different second components. Thus, the relation isa function.

b. Figure 1.15(b) shows that 6 corresponds to both 1 and 2. If any element in thedomain corresponds to more than one element in the range, the relation is nota function. This relation is not a function; two ordered pairs have the same firstcomponent and different second components.

Look at Figure 1.15(a) again. The fact that 1 and 2 in the domain correspondto the same number, 6, in the range does not violate the definition of a function.A function can have two different first components with the same second component.By contrast, a relation is not a function when two different ordered pairs have thesame first component and different second components. Thus, the relation inFigure 1.15(b) is not a function.

Check Point 2 Determine whether each relation is a function:

a. b.

Functions as EquationsFunctions are usually given in terms of equations rather than as sets of ordered pairs.For example, here is an equation that models the percentage of first-year collegewomen claiming no religious affiliation as a function of time:

The variable represents the number of years after 1970. The variable representsthe percentage of first-year college women claiming no religious affiliation. Thevariable is a function of the variable For each value of there is one and onlyone value of The variable is called the independent variable because it can beassigned any value from the domain. Thus, can be assigned any nonnegativeinteger representing the number of years after 1970. The variable is called thedependent variable because its value depends on The percentage claiming noreligious affiliation depends on the number of years after 1970. The value of thedependent variable, is calculated after selecting a value for the independentvariable,

We have seen that not every set of ordered pairs defines a function. Similarly,not all equations with the variables and define functions. If an equation is solvedfor and more than one value of can be obtained for a given then the equationdoes not define as a function of x.y

x,yyyx

x.y,

x.y

xxy.

x,x.y

yx

y = 0.013x2 - 0.21x + 8.7.

511, 22, 13, 42, 16, 52, 18, 526.511, 22, 13, 42, 15, 62, 15, 826

(6, 1) (6, 2)

Same first component

Different second components

516, 12, 16, 22, 18, 32, 19, 426.511, 62, 12, 62, 13, 82, 14, 926EXAMPLE 2

4321

986

Domain Range

Figure 1.15(b)

4321

986

Domain Range

Figure 1.15(a)

! Determine whether an equationrepresents a function.

Study TipIf a relation is a function, reversingthe components in each of its orderedpairs may result in a relation that isnot a function.


P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 150

Suppose we are interested in finding the function’s output when the input is30. To find the value of the function at 30, we substitute 30 for . We are evaluatingthe function at 30.

This is the given function.

Replace each occurrence of with 30.

Evaluate the exponential expression:

Perform the multiplications.

Subtract and add from left to right.

The statement read “ of 30 equals 14.1,” tells us that the value of thefunction at 30 is 14.1. When the function’s input is 30, its output is 14.1. Figure 1.17illustrates the input and output in terms of a function machine.

We have seen that in 2000, 13.2% actually claimed nonaffiliation, so our functionthat models the data slightly overestimates the percent for 2000.

f(30) =14.1

30 years after1970, or in 2000,

14.1% of first-year college womenclaimed no religious affiliation.

ff1302 = 14.1,

f1302 = 14.1 = 11.7 - 6.3 + 8.7

302 = 30 # 30 = 900. = 0.01319002 - 0.211302 + 8.7

x f1302 = 0.01313022 - 0.211302 + 8.7 f1x2 = 0.013x2 - 0.21x + 8.7

xf1302,

Inputx ! 30

Outputf(30) ! 14.1

f(x ) ! 0.013x 2 " 0.21x # 8.7

0.013(30) 2 " 0.21(30) # 8.7

Figure 1.17 A function machineat work

TechnologyGraphing utilities can be used to evaluate functions. The screens below show the evaluation of

at 30 on a TI-84 Plus graphing calculator. The function is named Y1.f

f1x2 = 0.013x2 - 0.21x + 8.7

We used to find To find other functionvalues, such as or substitute the specified input value, 40 or 55, for inthe function’s equation.

If a function is named and represents the independent variable, the notationcorresponds to the for a given Thus,

define the same function. This function may be written as

Evaluating a Function

If evaluate each of the following:

a. b. c.

Solution We substitute 2, and for in the equation for Whenreplacing with a variable or an algebraic expression, you might find it helpful tothink of the function’s equation as

f( x ) = x 2 + 3 x + 5.

xf.x-xx + 3,

f1-x2.f1x + 32f122f1x2 = x2 + 3x + 5,

EXAMPLE 4

y = f1x2 = 0.013x2 - 0.21x + 8.7.

f1x2 = 0.013x2 - 0.21x + 8.7� and� y = 0.013x2 - 0.21x + 8.7

x.y-valuef1x2 xf

xf1552,f1402 f1302.f1x2 = 0.013x2 - 0.21x + 8.7


P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 152

Use the graph of each function to identify its domain and its range.


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

• A closed dot indicates that the graph does not extend beyond this point andthe point belongs to the graph.

• An open dot indicates that the graph does not extend beyond this point andthe point does not belong to the graph.

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

y

x

y

x

y

x

Solution is a function of for the graphs in (b) and (c).xy

x

y

is not a function ofTwo values of correspond to an x-value.

yx.y

x

y

is a function of x.y

x

y

is a function of x.y

x

y

is not a function ofTwo values of correspond to an x-value.

yx.y

a. b. c. d.

Check Point 6 Use the vertical line test to identify graphs in which is afunction of

a. b. c.x.

y

! Obtain information about a function from its graph.

Analyzing the Graph of a Function

The human immunodeficiency virus, or HIV, infects andkills helper T cells. Because T cells stimulate the immunesystem to produce antibodies, their destruction disables thebody’s defenses against other pathogens. By counting thenumber of T cells that remain active in the body, theprogression of HIV can be monitored. The fewer helper Tcells, the more advanced the disease. Figure 1.20 shows agraph that is used to monitor the average progression of thedisease. The average number of T cells, is a function oftime after infection,

a. Explain why represents the graph of a function.b. Use the graph to find c. For what value of is d. Describe the general trend shown by the graph.

f1x2 = 350?x

f182.f

x.f(x),

EXAMPLE 7

11Time after Infection (years)

x1 2 3 4 5 6 7 8 9 10

1200

1000

800

600

400

200Ave

rage

T C

ell C

ount

(per

mill

ilite

r of

blo

od)

T Cell Count and HIV Infectiony

Asymptomatic StageFew or no symptoms present.

AIDS ClinicalDiagnosis

Symptomatic StageSymptoms begin or

get worse.

y � f(x)

Figure 1.20Source: B. E. Pruitt et al., Human Sexuality , Prentice Hall, 2007

Solution

a. No vertical line can be drawn that intersects the graph of more than once. By the vertical line test, represents thegraph of a function.

ff

P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 155


Figure 1.24 illustrates how the graph of afunction is used to determine the function’sdomain and its range.

Let’s apply these ideas to the graph of the function shown in Figure 1.25. Tofind the domain, look for all the inputs on the that correspond to points onthe graph. Can you see that they extend from to 2, inclusive? The function’sdomain can be represented as follows:

-4x-axis

Found on the x -axis

Domain: set of inputs

Found on the y -axis

Range: set of outputs

y � f (x )

x

y

0Domain

Range

Figure 1.24 Domain and range of f

Check Point 7a. Use the graph of in Figure 1.20 on page 155 to find b. For what value of is c. Estimate the minimum average T cell count during the asymptomatic stage.

Identifying Domain and Range from a Function’s Graph

f1x2 = 100?x

f152.f

Study TipThroughout this discussion, we will be using interval notation. Recall that square bracketsindicate endpoints that are included in an interval. Parentheses indicate endpoints that arenot included in an interval. Parentheses are always used with or For more detail oninterval notation, see Section P.9, pages 115–116.

- q .q

! Identify the domain and range of a function from its graph.

x

Domain: Inputs onx -axis extend from 4 to 2, inclusive.

Range: Outputson y -axis extendfrom 1 to 4, inclusive.y � f (x )

1 2 3 4 5 1

123

5

2 3 4 5

1 2 3 4 5

y

Figure 1.25 Domain and range of f

Using Set-Builder Notation Using Interval Notation

The setof all x

suchthat

x is greater than or equal to 4 and less than or equal to 2.

The square brackets indicate 4and 2 are included. Note the square

brackets on the x -axis in Figure 1.25.

{ x | –4 ! x ! 2 } [–4, 2].

To find the range, look for all the outputs on the that correspond topoints on the graph. They extend from 1 to 4, inclusive. The function’s range can berepresented as follows:

y-axis

Using Set-Builder Notation Using Interval Notation

The square brackets indicate 1and 4 are included. Note the square

brackets on the y -axis in Figure 1.25.

[1, 4].

The setof all y

suchthat

y is greater than or equal to1 and less than or equal to 4.

{ y | 1 ! y ! 4 }

Identifying the Domain and Range of a Functionfrom Its Graph

Use the graph of each function shown on the next page to identify its domain andits range.

EXAMPLE 8

P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 157

a .

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

y � f (x )

b.

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

y � f (x )

c.

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

y � f (x )

d.

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

y � f (x )

e.

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

y � f (x )

Solution For the graph of each function, the domain is highlighted in purple onthe and the range is highlighted in green on the y-axis.x-axis

a.

or or 30, 34 Range = 5y ƒ 0 … y … 36 3-2, 14 Domain = 5x ƒ -2 … x … 16

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

Range: Outputs on y -axis extend from0 to 3, inclusive.


y � f (x )

b.

or or 11, 24 Range = 5y ƒ 1 6 y … 26 1-3, 24 Domain = 5x ƒ -3 6 x … 26

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

Domain: Inputs onx -axis extend from 3, excluding 3,to 2, including 2.

Range: Outputs on y -axis extend from1, excluding 1, to 2,including 2.

y � f (x )

c.

or or 31, 5 4 Range = 5y ƒ 1 … y … 5 6 3-2, 12 Domain = 5x ƒ -2 … x 6 16

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

Domain: Inputs onx -axis extend from 2, including 2,to 1, excluding 1.


y � f (x )

d.

or or 30, q2 Range = 5y ƒ y Ú 06 1- q , 44 Domain = 5x ƒ x … 46

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

Domain: Inputs onx -axis include realnumbers less than

or equal to 4.

Range: Outputs on y -axis include realnumbers greater

than or equal to O.

x

y � f (x )

e.

or Range = 5y ƒ y = 1, 2, 36 31, 42 Domain = 5x ƒ 1 … x 6 46

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

Domain: Inputs onx -axis extend from1, including 1, to4, excluding 4.

Range: Outputson y -axis "step"from 1 to 2 to 3.

y � f (x )


P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 158

a .

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

y � f (x )

b.

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

y � f (x )

c.

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

y � f (x )

d.

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

y � f (x )

e.

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

y � f (x )

Solution For the graph of each function, the domain is highlighted in purple onthe and the range is highlighted in green on the y-axis.x-axis

a.

or or 30, 34 Range = 5y ƒ 0 … y … 36 3-2, 14 Domain = 5x ƒ -2 … x … 16

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5



y � f (x )

b.

or or 11, 24 Range = 5y ƒ 1 6 y … 26 1-3, 24 Domain = 5x ƒ -3 6 x … 26

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

Domain: Inputs onx -axis extend from 3, excluding 3,to 2, including 2.

Range: Outputs on y -axis extend from1, excluding 1, to 2,including 2.

y � f (x )

c.

or or 31, 5 4 Range = 5y ƒ 1 … y … 5 6 3-2, 12 Domain = 5x ƒ -2 … x 6 16

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

Domain: Inputs onx -axis extend from 2, including 2,to 1, excluding 1.


y � f (x )

d.

or or 30, q2 Range = 5y ƒ y Ú 06 1- q , 44 Domain = 5x ƒ x … 46

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

Domain: Inputs onx -axis include realnumbers less than

or equal to 4.

Range: Outputs on y -axis include realnumbers greater

than or equal to O.

x

y � f (x )

e.

or Range = 5y ƒ y = 1, 2, 36 31, 42 Domain = 5x ƒ 1 … x 6 46

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

Domain: Inputs onx -axis extend from1, including 1, to4, excluding 4.

Range: Outputson y -axis "step"from 1 to 2 to 3.

y � f (x )


P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 158

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1.2 Basics of Functions and Their Graphs Notes