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E. Applications of Rational Functions

E. Applications of Rational Functions

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CHAPTER 4 Polynomial and Rational Functions



4.4 EXERCISES





CONCEPTS AND VOCABULARY



Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.



1. Write the following in direction/approach notation.

As x becomes an infinitely large negative number, y

approaches 2.



2. For any constant k, the notation as ͿxͿ S ϩq,

y S k is an indication of a

asymptote,

while x S k, ͿyͿ S ϩq indicates a

asymptote.



3. Vertical asymptotes are found by setting the

equal to zero. The x-intercepts are found

by setting the

equal to zero.



4. If the degree of the numerator is equal to the

degree of the denominator, a horizontal asymptote

occurs at y ϭ ab , where ab represents the ratio of

the

.



3x2 Ϫ 2x

and a table of

2x2 Ϫ 3

values to discuss the concept of horizontal

asymptotes. At what positive value of x is the graph

of g within 0.01 of its horizontal asymptote?



5. Use the function g1x2 ϭ







6. Name all of the “tools” at your disposal that play a

role in the graphing of rational functions. Which

tools are indispensable and always used? Which are

used only as the situation merits?



DEVELOPING YOUR SKILLS



Give the location of the vertical asymptote(s) if they

exist, and state the function’s domain.



7. f 1x2 ϭ



xϩ2

xϪ3



9. g1x2 ϭ



3x2

x2 Ϫ 9



8. F1x2 ϭ



4x

2x Ϫ 3



10. G1x2 ϭ



xϩ1

9x2 Ϫ 4



11. h1x2 ϭ



x2 Ϫ 1

xϪ5

12. H1x2 ϭ 2

2x ϩ 3x Ϫ 5

2x Ϫ x Ϫ 3



13. p1x2 ϭ



2x ϩ 3

2

x ϩxϩ1



2



14. q1x2 ϭ



2x3

x2 ϩ 4



Give the location of the vertical asymptote(s) if they

exist, and state whether function values will change sign

(positive to negative or negative to positive) from one

side of the asymptote to the other.



15. Y1 ϭ



xϩ1

x ϪxϪ6

2



17. r1x2 ϭ



16. Y2 ϭ



2x ϩ 3

x Ϫ x Ϫ 20

2



x2 ϩ 3x Ϫ 10

x2 Ϫ 2x Ϫ 15

18.

R1x2

ϭ

x2 Ϫ 6x ϩ 9

x2 Ϫ 4x ϩ 4



19. Y1 ϭ



x

x ϩ 2x Ϫ 4x Ϫ 8



20. Y2 ϭ



Ϫ2x

x3 ϩ x2 Ϫ x Ϫ 1



3



2



For the functions given, (a) determine if a horizontal

asymptote exists and (b) determine if the graph will

cross the asymptote, and if so, where it crosses.



21. Y1 ϭ



2x Ϫ 3

x2 ϩ 1



22. Y2 ϭ



4x ϩ 3

2x2 ϩ 5



23. r1x2 ϭ



4x2 Ϫ 9

2x2 Ϫ x Ϫ 10

24.

R1x2

ϭ

x2 Ϫ 3x Ϫ 18

x2 ϩ 5



25. p1x2 ϭ



3x2 Ϫ 5

x2 Ϫ 1



26. P1x2 ϭ



3x2 Ϫ 5x Ϫ 2

x2 Ϫ 4



Apply long division to find the quotient and remainder

for each function. Use this information to determine the

equation of the horizontal asymptote, and whether the

graph will cross this asymptote. Verify answers by

graphing the functions on a graphing calculator and

locating points of intersection.



27. v1x2 ϭ



8x

x ϩ1



28. f 1x2 ϭ



4x ϩ 8

x2 ϩ 1



29. g1x2 ϭ



2x2 Ϫ 8x

x2 Ϫ 4



30. h1x2 ϭ



x2 Ϫ x Ϫ 6

x2 Ϫ 1



2



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Give the location of the x- and y-intercepts (if they

exist), and discuss the behavior of the function (bounce

or cross) at each x-intercept.



x2 Ϫ 3x

31. f 1x2 ϭ 2

x Ϫ5



2x Ϫ x2

32. F1x2 ϭ 2

x ϩ 2x Ϫ 3



33. g1x2 ϭ



x2 ϩ 3x Ϫ 4

x2 ϩ 7x ϩ 6

34.

G1x2

ϭ

x2 Ϫ 1

x2 Ϫ 2



35. h1x2 ϭ



x3 Ϫ 6x2 ϩ 9x

4x ϩ 4x2 ϩ x3

36. H1x2 ϭ

2

4Ϫx

x2 Ϫ 1



Use the Guidelines for Graphing Rational Functions to

graph the functions given.



xϩ3

37. f 1x2 ϭ

xϪ1

39. F1x2 ϭ



8x

x ϩ4

2



Ϫ2x

41. p1x2 ϭ 2

x Ϫ4

2



40. G1x2 ϭ



Ϫ12x

x2 ϩ 3



Ϫ3x

2x

46. H1x2 ϭ 2

x2 Ϫ 6x ϩ 9

x Ϫ 2x ϩ 1



xϪ1

x Ϫ 3x Ϫ 4



49. s1x2 ϭ



4x2

2x2 ϩ 4



x2 Ϫ 4

51. Y1 ϭ 2

x Ϫ1



54. V1x2 ϭ



3x

x ϩx ϪxϪ1



48. Y2 ϭ



1Ϫx

x2 Ϫ 2x



50. S1x2 ϭ



Ϫ2x2

x2 ϩ 1



x2 Ϫ x Ϫ 12

52. Y2 ϭ 2

x ϩ x Ϫ 12



3



3



2



Use the vertical asymptotes, x-intercepts, and their

multiplicities to construct an equation that corresponds

to each graph. Be sure the y-intercept estimated from

the graph matches the value given by your equation for

x ‫ ؍‬0. Check work on a graphing calculator.



55.



56.



y

10

8

6

4

2

Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2

Ϫ2



57.



2 4 6 8 10 x



Ϫ4

Ϫ6

Ϫ8

Ϫ10



58.



y



Ϫ4

Ϫ6

Ϫ8

Ϫ10



10

8

6

4

2



Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2

Ϫ2



2 4 6 8 10 x



10

8

6

4

2

Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2

Ϫ2



y



(Ϫ2, 2)



(1, 1)



Ϫ4

Ϫ6

Ϫ8

Ϫ10



3x

42. P1x2 ϭ 2

x Ϫ9



45. h1x2 ϭ



2



Ϫ2x

x ϩ 2x2 Ϫ 4x Ϫ 8



2



2x Ϫ x2

x2 ϩ 3x

44.

Q1x2

ϭ

x2 ϩ 4x Ϫ 5

x2 Ϫ 2x Ϫ 3



47. Y1 ϭ



53. v1x2 ϭ



xϪ4

38. g1x2 ϭ

xϩ2



43. q1x2 ϭ







441



Section 4.4 Graphing Rational Functions



4 6 8 10 x



΂5, Ϫ Ò

16 ΃



y

10

8

6

4

2

Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2

Ϫ2



(3, 4)

2 4 6 8 10 x



Ϫ4

Ϫ6

Ϫ8

Ϫ10



Graph the following using the Guidelines for Graphing

Rational Functions. From the equations given, note

there are no horizontal asymptotes (the degree of each

numerator is greater than the degree of the

denominator) and a large number of plotted points may

be necessary to complete each graph.



59. v1x2 ϭ



x2 Ϫ 4

x



60. f 1x2 ϭ



9 Ϫ x2

xϪ1



61. g1x2 ϭ



x2

xϪ1



62. h1x2 ϭ



1 Ϫ x2

xϩ2



WORKING WITH FORMULAS

ax

x ؉b

The population density of urban areas (in people

per square mile) can be modeled by the formula

shown, where a and b are constants related to the

overall population and sprawl of the area under

study, and D(x) is the population density (in

hundreds), x mi from the center of downtown.



63. Population density: D1x2 ‫؍‬



2



Graph the function for a ϭ 63 and b ϭ 20 over the

interval x ʦ 3 0, 504 , and then use the graph to

answer the following questions.



a. What is the significance of the horizontal

asymptote (what does it mean in this context)?

b. How far from downtown does the population

density fall below 525 people per square mile?

How far until the density falls below 300

people per square mile?

c. Use the graph and a table to determine how far

from downtown the population density reaches

a maximum. What is this maximum?



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kx

100 ؊ x

Some industries resist cleaner air standards because

the cost of removing pollutants rises dramatically as

higher standards are set. This phenomenon can be

modeled by the formula given, where C(x) is the

cost (in thousands of dollars) of removing x% of the

pollutant and k is a constant that depends on the type

of pollutant and other factors.

Graph the function for k ϭ 250 over the interval

x ʦ 3 0, 1004 , and then use the graph to answer the

following questions.



4–62



64. Cost of removing pollutants: C1x2 ‫؍‬







a. What is the significance of the vertical

asymptote (what does it mean in this context)?

b. If new laws are passed that require 80% of a

pollutant to be removed, while the existing law

requires only 75%, how much will the new

legislation cost the company? Compare the

cost of the 5% increase from 75% to 80% with

the cost of the 1% increase from 90% to 91%.

c. What percent of the pollutants can be removed

if the company budgets 2250 thousand dollars?



APPLICATIONS



C(h)

65. Medication in the blood0.4

stream: The concentration C 0.3

0.2

of a certain medicine in the

0.1

bloodstream h hours after

2 6 10 14 18 22 26 h

being injected into the

shoulder is given by the

2h2 ϩ h

. Use the given graph of

function: C1h2 ϭ 3

h ϩ 70

the function to answer the following questions.

a. Approximately how many hours after injection

did the maximum concentration occur? What

was the maximum concentration?

b. Use C(h) to compute the rate of change for the

intervals h ϭ 8 to h ϭ 10 and h ϭ 20 to

h ϭ 22. What do you notice?

c. Use mathematical notation to state what

happens to the concentration C as the number of

hours becomes infinitely large. What role does

the h-axis play for this function?



66. Supply and demand: In

S(t)

10.0

response to certain market

7.5

demands, manufacturers will 5.0

2.5

quickly get a product out on

the market to take advantage

10 20 30 40 50 60 70 t

of consumer interest. Once

the product is released, it is not uncommon for

sales to initially skyrocket, taper off, and then

gradually decrease as consumer interest wanes. For

a certain product, sales can be modeled by the

250t

, where S(t) represents the

function S1t2 ϭ 2

t ϩ 150

daily sales (in $10,000) t days after the product has

debuted. Use the given graph of the function to

answer the following questions.

a. Approximately how many days after the

product came out did sales reach a maximum?

What was the maximum sales?



b. Use S(t) to compute the rate of change for the

intervals t ϭ 7 to t ϭ 8 and t ϭ 60 to t ϭ 62.

What do you notice?

c. Use mathematical notation to state what

happens to the daily sales S as the number of

days becomes infinitely large. What role does

the t-axis play for this function?

67. Cost to remove pollutants: For a certain coalburning power plant, the cost to remove pollutants

from plant emissions can be modeled by

80p

C1p2 ϭ

, where C(p) represents the cost

100 Ϫ p

(in thousands of dollars) to remove p percent of the

pollutants. (a) Find the cost to remove 20%, 50%,

and 80% of the pollutants, then comment on the

results; (b) graph the function using an appropriate

scale; and (c) use mathematical notation to state

what happens if the power company attempts to

remove 100% of the pollutants.

68. Costs of recycling: A large city has initiated a new

recycling effort, and wants to distribute recycling bins

for use in separating various recyclable materials.

City planners anticipate the cost of the program can

220p

, where

be modeled by the function C1p2 ϭ

100 Ϫ p

C(p) represents the cost (in $10,000) to distribute

the bins to p percent of the population. (a) Find the

cost to distribute bins to 25%, 50%, and 75% of the

population, then comment on the results; (b) graph

the function using an appropriate scale; and (c) use

mathematical notation to state what happens if the

city attempts to give recycling bins to 100% of the

population.



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Memory retention: Due to their asymptotic behavior,

rational functions are often used to model the mind’s ability

to retain information over a long period of time — the “use

it or lose it” phenomenon.



69. Language retention: A large group of students is

asked to memorize a list of 50 Italian words, a

language that is unfamiliar to them. The group is

then tested regularly to see how many of the words

are retained over a period of time. The average

number of words retained is modeled by the

6t ϩ 40

function W1t2 ϭ

, where W(t) represents

t

the number of words remembered after t days.

a. Graph the function over the interval t ʦ 3 1, 404 .

How many days until only half the words are

remembered? How many days until only onefifth of the words are remembered?

b. After 10 days, what is the average number of

words retained? How many days until only

8 words can be recalled?

c. What is the significance of the horizontal

asymptote (what does it mean in this

context)?

70. Language retention: A similar study asked

students to memorize 50 Hawaiian words, a

language that is both unfamiliar and phonetically

foreign to them (see Exercise 69). The average

number of words retained is modeled by the

4t ϩ 20

, where W(t) represents

function W1t2 ϭ

t

the number of words after t days.

a. Graph the function over the interval t ʦ 31, 40 4.

How many days until only half the words are

remembered? How does this compare to

Exercise 69? How many days until only onefifth of the words are remembered?

b. After 7 days, what is the average number of

words retained? How many days until only

5 words can be recalled?

c. What is the significance of the horizontal

asymptote (what does it mean in this

context)?

Concentration and dilution: When

antifreeze is mixed with water, it

becomes diluted — less than 100%

antifreeze. The more water added,

the less concentrated the antifreeze

becomes, with this process continuing until a desired

concentration is met. This application and many similar to

it can be modeled by rational functions.



Section 4.4 Graphing Rational Functions



443



71. Concentration of antifreeze: A 400-gal tank

currently holds 40 gal of a 25% antifreeze

solution. To raise the concentration of the

antifreeze in the tank, x gal of a 75% antifreeze

solution is pumped in.

a. Show the formula for the resulting

40 ϩ 3x

concentration is C1x2 ϭ

after

160 ϩ 4x

simplifying, and graph the function over the

interval x ʦ 30, 360 4 .

b. What is the concentration of the antifreeze in the

tank after 10 gal of the new solution are added?

After 120 gal have been added? How much

liquid is now in the tank?

c. If the concentration level is now at 65%, how

many gallons of the 75% solution have been

added? How many gallons of liquid are in the

tank now?

d. What is the maximum antifreeze concentration

that can be attained in a tank of this size? What

is the maximum concentration that can be

attained in a tank of “unlimited” size?

72. Concentration of sodium chloride: A sodium

chloride solution has a concentration of 0.2 oz

(weight) per gallon. The solution is pumped into an

800-gal tank currently holding 40 gal of pure

water, at a rate of 10 gal/min.

a. Find a function A(t) modeling the amount of

liquid in the tank after t min, and a function

S(t) for the amount of sodium chloride in the

tank after t min.

b. The concentration C(t) in ounces per gallon is

S1t2

, a rational function.

measured by the ratio

A1t2

Graph the function on the interval t ʦ 30, 1004.

What is the concentration level (in ounces per

gallon) after 6 min? After 28 min? How many

gallons of liquid are in the tank at this time?

c. If the concentration level is now 0.184 oz/gal,

how long have the pumps been running?

How many gallons of liquid are in the tank

now?

d. What is the maximum concentration that can

be attained in a tank of this size? What is the

maximum concentration that can be attained in

a tank of “unlimited” size?



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Average cost of manufacturing an item: The cost “C” to

manufacture an item depends on the relatively fixed costs

“K” for remaining in business (utilities, maintenance,

transportation, etc.) and the actual cost “c” of

manufacturing the item (labor and materials). For x items

the cost is C1x2 ϭ K ϩ cx. The average cost

C1x2

“A” of manufacturing an item is then A1x2 ϭ

.

x



73. Manufacturing water heaters: A company that

manufactures water heaters finds their fixed costs

are normally $50,000 per month, while the cost to

manufacture each heater is $125. Due to factory

size and the current equipment, the company can

produce a maximum of 5000 water heaters per

month during a good month.

a. Use the average cost function to find the

average cost if 500 water heaters are

manufactured each month. What is the average

cost if 1000 heaters are made?

b. What level of production will bring the average

cost down to $150 per water heater?

c. If the average cost is currently $137.50, how

many water heaters are being produced that

month?

d. What’s the significance of the horizontal

asymptote for the average cost function (what

does it mean in this context)? Will the

company ever break the $130 average cost

level? Why or why not?

74. Producing biodegradable disposable diapers:

An enterprising company has finally developed a

better disposable diaper that is biodegradable. The

brand becomes wildly popular and production is

soaring. The fixed cost of production is $20,000

per month, while the cost of manufacturing is

$6.00 per case (48 diapers). Even while working

three shifts around-the-clock, the maximum

production level is 16,000 cases per month. The

company figures it will be profitable if it can bring

costs down to an average of $7 per case.

a. Use the average cost function to find the

average cost if 2000 cases are produced each

month. What is the average cost if 4000 cases

are made?

b. What level of production will bring the average

cost down to $8 per case?

c. If the average cost is currently $10 per case,

how many cases are being produced?

d. What’s the significance of the horizontal

asymptote for the average cost function (what

does it mean in this context)? Will the

company ever reach its goal of $7/case at its

maximum production? What level of

production would help them meet their goal?



4–64

Test averages and grade point

averages: To calculate a test

average we sum all test points P

and divide by the number of

P

tests N: . To compute the

N

score or scores needed on future tests to raise the average

grade to a desired grade G, we add the number of additional

tests n to the denominator, and the number of additional

tests times the projected grade g on each test to the

numerator:

P ϩ ng

G1n2 ϭ

. The result is a rational function with

Nϩn

some “eye-opening” results.



75. Computing an average grade: After four tests,

Bobby Lou’s test average was an 84.

[Hint: P ϭ 41842 ϭ 336.]

a. Assume that she gets a 95 on all remaining

tests 1g ϭ 952. Graph the resulting function on

a calculator using the window n ʦ 30, 20 4 and

G1n2 ʦ 380 to 100 4 . Use the calculator to

determine how many tests are required to lift

her grade to a 90 under these conditions.

b. At some colleges, the range for an “A” grade is

93–100. How many tests would Bobby Lou

have to score a 95 on, to raise her average to

higher than 93? Were you surprised?

c. Describe the significance of the horizontal

asymptote of the average grade function. Is a

test average of 95 possible for her under these

conditions?

d. Assume now that Bobby Lou scores 100 on all

remaining tests 1g ϭ 1002. Approximately how

many more tests are required to lift her grade

average to higher than 93?

76. Computing a GPA: At most colleges, A S 4

grade points, B S 3, C S 2, and D S 1. After

taking 56 credit hours, Aurelio’s GPA is 2.5.

[Hint: In the formula given, P ϭ 2.51562 ϭ 140.]

a. Assume Aurelio is determined to get A’s

(4 grade points or g ϭ 42, for all remaining

credit hours. Graph the resulting function on a

calculator using the window n ʦ 3 0, 604 and

G1n2 ʦ 3 2, 4 4. Use the calculator to determine

the number of credit hours required to lift his

GPA to over 2.75 under these conditions.

b. At some colleges, scholarship money is

available only to students with a 3.0 average or

higher. How many (perfect 4.0) credit hours

would Aurelio have to earn, to raise his GPA to

3.0 or higher? Were you surprised?

c. Describe the significance of the horizontal

asymptote of the GPA function. Is a GPA of

4.0 possible for him under these conditions?



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Section 4.5 Additional Insights into Rational Functions



EXTENDING THE CONCEPT



77. In addition to determining if a function has a

vertical asymptote, we are often interested in how

fast the graph approaches the asymptote. As in

previous investigations, this involves the function’s

rate of change over a small interval. Exercise 64

describes the rising cost of removing pollutants

from the air. As noted there, the rate of increase in

the cost changes as higher requirements are set. To

quantify this change, we’ll compute the rate of

C1x2 2 Ϫ C1x1 2

¢C

250x

ϭ

.

change

for C1x2 ϭ

x2 Ϫ x1

¢x

100 Ϫ x

a. Find the rate of change of the function in the

following intervals:

x ʦ 3 60, 614 x ʦ 3 70, 71 4

x ʦ 3 80, 814 x ʦ 390, 91 4

b. What do you notice? How much did the rate

increase from the first interval to the second?

From the second to the third? From the third to

the fourth?







445



c. Recompute parts (a) and (b) using the function

350x

C1x2 ϭ

. Comment on what you notice.

100 Ϫ x

ax2 ϩ k

, where a, b,

bx2 ϩ h

k, and h are constants and a, b 7 0.

a. What can you say about asymptotes and

intercepts of this function if h, k 7 0?

b. Now assume k 6 0 and h 7 0. How does this

affect the asymptotes? The intercepts?

c. If b ϭ 1 and a 7 1, how does this affect the

results from part (b)?

d. How is the graph affected if k 7 0 and h 6 0?

e. Find values of a, b, h, and k that create a

function with a horizontal asymptote at y ϭ 32,

x-intercepts at 1Ϫ2, 02 and (2, 0), a y-intercept

of 10, Ϫ42 , and no vertical asymptotes.



78. Consider the function f 1x2 ϭ



MAINTAINING YOUR SKILLS



79. (1.4) Find the equation of a line that is

perpendicular to 3x Ϫ 4y ϭ 12 and contains the

point 12, Ϫ32.



80. (4.1) Use synthetic division and the remainder

theorem to find the value of f (4), f 1 32 2 , and

f (2):f 1x2 ϭ 2x3 Ϫ 7x2 ϩ 5x ϩ 3.



81. (3.2) Solve the following equation using the

quadratic formula, then write the equation in

factored form: 12x2 ϩ 55x Ϫ 48 ϭ 0.



82. (R.1/3.1) Describe/Define each set of numbers:

complex ‫ރ‬, rational ‫ޑ‬, and integers ‫ޚ‬.



4.5



Additional Insights into Rational Functions



LEARNING OBJECTIVES

In Section 4.5 you will see

how we can:



A. Graph rational functions with

removable discontinuities

B. Graph rational functions

with oblique or nonlinear

asymptotes

C. Solve applications involving

rational functions



In Section 4.4, we studied rational functions whose graphs had horizontal and/or vertical asymptotes. In this section, we’ll study functions with asymptotes that are neither

horizontal nor vertical. In addition, we’ll further explore the “break” we saw in graphs

of certain piecewise-defined functions, that of a simple “hole” created when the numerator and denominator of a rational function share a common variable factor.



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CHAPTER 4 Polynomial and Rational Functions



A. Rational Functions and Removable Discontinuities

In Example 6 of Section 2.5, we graphed the piecewise-defined function

x2 Ϫ 4

x 2

h1x2 ϭ • x Ϫ 2

. The second piece is simply the point (2, 1). The first piece

1

xϭ2

is a rational function, but instead of a vertical asymptote at x ϭ 2 (the zero of the

denominator), its graph is actually the line y ϭ x ϩ 2 with a “hole” at (2, 4) called a

removable discontinuity (Figure 4.60). The hole occurs because the numerator and

x2 Ϫ 4

denominator of y ϭ

share the common factor 1x Ϫ 22 , and canceling these

xϪ2

factors leaves y ϭ x ϩ 2, a continuous function. However, the original function is not

defined at x ϭ 2, so we must delete the single point at x ϭ 2, y ϭ 4 from the graph of

the line (Figure 4.60).

Figure 4.60



Figure 4.61



y



y



5



5



4



4



(2, 4)



3

2



h(x)



2



H(x)



(2, 1)



1



Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

Ϫ1



(2, 4)



3



1



2



3



4



5



1



Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

Ϫ1



x



Ϫ2



Ϫ2



Ϫ3



Ϫ3



Ϫ4



Ϫ4



Ϫ5



Ϫ5



1



2



3



4



5



x



We can remove or fix this discontinuity by redefining the second piece

as h1x2 ϭ 4, when x ϭ 2. This would create a new and continuous function,

x2 Ϫ 4

x 2

H1x2 ϭ • x Ϫ 2

(Figure 4.61).

4

xϭ2

It’s possible for a rational graph to have more than one removable discontinuity, or

to be nonlinear with a removable discontinuity. For cases where we elect to repair the

break, we will adopt the convention of using the corresponding upper case letter to

name the new function, as we did here.

EXAMPLE 1







Graphing Rational Functions with Removable Discontinuities

x3 ϩ 8

. If there is a removable discontinuity, repair the

xϩ2

break using an appropriate piecewise-defined function.

Graph the function t1x2 ϭ



Solution







Note the domain of t does not include x ϭ Ϫ2. We begin by factoring as before to

identify zeroes and asymptotes, but find the numerator and denominator share a

common factor, which we remove.

t1x2 ϭ

ϭ



x3 ϩ 8

xϩ2

1x ϩ 221x2 Ϫ 2x ϩ 42



xϩ2

ϭ x Ϫ 2x ϩ 4; where x

2



Ϫ2



The graph of t will be the same as y ϭ x Ϫ 2x ϩ 4 for all values except x ϭ Ϫ2.

Here we have a parabola, opening upward, with y-intercept (0, 4). From the vertex

Ϫ1Ϫ22

Ϫb

ϭ 1, giving y ϭ 3

formula, the x-coordinate of the vertex will be

ϭ

2a

2112

2



cob19545_ch04_446-458.qxd



8/20/10



12:04 AM



Page 447



College Algebra Graphs & Models—



4–67



447



Section 4.5 Additional Insights into Rational Functions



after substitution. The vertex is (1, 3). Evaluating

t1Ϫ12 we find 1Ϫ1, 72 is on the graph, giving the

point (3, 7) using the axis of symmetry. We draw

a parabola through these points, noting the

original function is not defined at Ϫ2, and there

will be a “hole” in the graph at 1Ϫ2, y2. The

value of y is found by substituting Ϫ2 for x in the

simplified form: 1Ϫ22 2 Ϫ 21Ϫ22 ϩ 4 ϭ 12. This

information produces the graph shown. We can

repair the break using the function

x3 ϩ 8

x Ϫ2

T1x2 ϭ • x ϩ 2

12

x ϭ Ϫ2



t(x)



(4, 12)

from

symmetry



(Ϫ2, 12)

10



(3, 7)

from

symmetry



(Ϫ1, 7)

5



(1, 3)



5



x



axis of

symmetry



Now try Exercises 7 through 18

The







feature of a graphing calculator is a wonderful tool for understanding the

f 1x2

, where

characteristics of a function. We’ll illustrate using the function h1x2 ϭ

g1x2

TRACE



f 1x2 ϭ x3 Ϫ 2x2 Ϫ 3x ϩ 6 and g1x2 ϭ x Ϫ 2 (similar to Example 1). Enter



A. You’ve just seen how

we can graph rational

functions with removable

discontinuities



the Y= screen as Y1 (Figure 4.62), then graph

the function using ZOOM 4:ZDecimal. Recall

this will allow the calculator to trace through

¢x intervals of 0.1. After pressing the TRACE

key, the cursor appears on the graph at the

y-intercept 10, Ϫ32 and its location is displayed

at the bottom of the screen. Note that there is a

“hole” in the graph in the first quadrant (Figure 4.63). We can walk the cursor along the

curve in either direction using the left arrow

and right arrow

keys to determine exactly

where this hole occurs. Walking the cursor to

the right, we note that no output is displayed

for x ϭ 2.

f 1x2

Next, simplify

and enter the result

g1x2

as Y2. Factoring the numerator by grouping

and reducing the common factors gives

1x2 Ϫ 32 1x Ϫ 22

f 1x2

ϭ

, so Y2 ϭ x2 Ϫ 3.

g1x2

1x Ϫ 22

Graphing both functions reveals that they are

identical, except that Y2 ϭ x2 Ϫ 3 covers the

hole left by Y1 using x ϭ 2, y ϭ 1. In other

words, Y1 is equivalent to Y2 except at x ϭ 2.

This can also be seen using the TABLE feature

of a calculator, which displays an error message

for Y1 when x ϭ 2 is input, but shows an output

of 1 for Y2 (Figure 4.64). The bottom line is—

the domain of h is all real numbers except x ϭ 2.



f 1x2



g1x2



on



Figure 4.62



Figure 4.63

3.1



Ϫ4.7



4.7



Ϫ3.1



Figure 4.64



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E. Applications of Rational Functions

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