POWERS,ROOTS, AND RADICALS

POWERS, ROOTS,AND RADICALSPOWERS, ROOTS,AND RADICALS

� How can you estimate the weight of a dinosaur?

398

http://www.classzone.com

APPLICATION: Dinosaurs

Scientists have determinedrelationships between the bonemeasurements and the heightsand weights of living animals.By applying the relationships todinosaur bones, scientists canestimate heights and weightsof these prehistoric animals.

Think & DiscussThe table below gives the femur circumference (inmillimeters) and estimated weight (in kilograms)for four different dinosaurs that walk on two feet.

1. Graph the ordered pairs ( femur circumference,estimated weight) from the table. Why can’tyou model the data with a linear function?

2. The femur circumference of a Hypacrosaurusaltispinus is about 400 millimeters. Estimate areasonable weight for this dinosaur. How didyou derive your estimate?

Learn More About ItYou will apply the relationship between femurcircumference and weight to a Tyrannosaurus rexin Exercise 64 on p. 442.

APPLICATION LINK Visit www.mcdougallittell.com for more information about dinosaurs.

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7

399

�

Femur circumference Estimated weight

103 50

201 310

348 1400

504 3800


400 Chapter 7

What’s the chapter about?Chapter 7 is about powers, roots, and radicals. In Chapter 7 you’ll learn

• how to use rational exponents and nth roots of numbers.

• how to perform operations with and find inverses of functions.

• how to graph radical functions and solve radical equations.

CHAPTER

7Study Guide

PREVIEW

Are you ready for the chapter?SKILL REVIEW Do these exercises to review key skills that you’ll apply in thischapter. See the given reference page if there is something you don’t understand.

Solve the equation for y. (Review Example 1, p. 26)

1. 3x º 2y = 12 2. x + �12�y = 5 3. x = 4y º 1

Factor the trinomial. (Review Examples 1 and 2, p. 256)

4. x2 + 10x + 21 5. x2 + 5x º 36 6. 2x2 º 16x + 30

Simplify the expression. (Review Example 2, p. 324)

7. (abc2)4 8. x5 • xº3 9. ��xy

2��2

10. �3yx� •

Perform the indicated operation. (Review Examples 1–6, pp. 338 and 339)

11. 5x2(x º 8) 12. (3y º 2)2 13. (7x2 + x) º (6x º 4)

3x2yº2�

12y3

PREPARE

Here’s a study strategy!STUDY

STRATEGY

� Review• exponent, p. 11

• relation, p. 67

• function, p. 67

• square root, p. 264

� New• n th root of a, p. 401

• power function, p. 415

• composition, p. 416

• inverse function, p. 422

• radical function, p. 431

• measure of central tendency,p. 445

• measure of dispersion, p. 446

• box-and-whisker plot, p. 447

• histogram, p. 448

• frequency distribution, p. 448

KEY VOCABULARY

Quiz Yourself

After you complete a homework assignment, copy afew representative problems from the assignment ona separate piece of paper. Record the lesson numberfor the problems and leave space for the answers.You can use these problems to quiz yourself later,such as before a class quiz is given.

STUDENT HELP

Study Tip“Student Help” boxesthroughout the chaptergive you study tips andtell you where to look forextra help in this bookand on the Internet.


7.1 nth Roots and Rational Exponents 401

nth Roots and RationalExponents

EVALUATING N TH ROOTS

You can extend the concept of a square root to other types of roots. For instance, 2 is a cube root of 8 because 23 = 8, and 3 is a fourth root of 81 because 34 = 81. In general, for an integer n greater than 1, if bn = a, then b is an An nth root of a is written as �n

a�, where n is the of the radical.You can also write an nth root of a as a power of a. For the particular case of a square root, suppose that �a� = ak. Then you can determine a value for k as follows:

�a� • �a� = a Definition of square root

ak • ak = a Substitute ak for �a�.

a2k = a1 Product of powers property

2k = 1 Set exponents equal when bases are equal.

k = �12� Solve for k.

Therefore, you can see that �a� = a1/ 2. In a similar way you can show that �3 a� = a1/3 and �4 a� = a1/4. In general, �n a� = a1/n for any integer n greater than 1.

Finding nth Roots

Find the indicated real nth root(s) of a.

a. n = 3, a = º125 b. n = 4, a = 16

SOLUTION

a. Because n = 3 is odd, a = º125 has one real cube root. Because (º5)3 = º125,you can write:

�3 º�1�2�5� = º5 or (º125)1/3 = º5

b. Because n = 4 is even and a = 16 > 0, 16 has two real fourth roots. Because24 = 16 and (º2)4 = 16, you can write:

±�4 1�6� = ±2 or ±161/4 = ±2

E X A M P L E 1

indexnth root of a.

GOAL 1

Evaluate n th rootsof real numbers using bothradical notation and rationalexponent notation.

Use n th roots tosolve real-life problems,such as finding the total massof a spacecraft that can besent to Mars in Example 5.

� To solve real-lifeproblems, such as finding the number of reptile andamphibian species thatPuerto Rico can support in Ex. 67.

Why you should learn it

GOAL 2

GOAL 1

What you should learn

7.1RE

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Let n be an integer greater than 1 and let a be a real number.

• If n is odd, then a has one real nth root: �n a� = a1/n

• If n is even and a > 0, then a has two real nth roots: ±�n a� = ±a1/n

• If n is even and a = 0, then a has one nth root: �n 0� = 01/n = 0

• If n is even and a < 0, then a has no real nth roots.

REAL NTH ROOTS


402 Chapter 7 Powers, Roots, and Radicals

A rational exponent does not have to be of the form �1n� where n is an integer greater

than 1. Other rational numbers such as �32� and º�

12� can also be used as exponents.

Evaluating Expressions with Rational Exponents

a. 93/2 = (�9�)3 = 33 = 27 Using radical notation

93/2 = (91/2)3 = 33 = 27 Using rational exponent notation

b. 32º2/5 = �32

12/5� = = �

212� = �

14� Using radical notation

32º2/5 = �32

12/5� = = �

212� = �

14� Using rational exponent notation

. . . . . . . . . .

When using a graphing calculator to approximate an nth root, you may have torewrite the nth root using a rational exponent. Then use the calculator’s power key.

Approximating a Root with a Calculator

Use a graphing calculator to approximate (�4 5�)3.

SOLUTION First rewrite (�4 5�)3 as 53/4. Then enter the following:

Keystrokes: 5 3 4 Display:

� (�4 5�)3 ≈ 3.34

. . . . . . . . . .

To solve simple equations involving xn, isolate the power and then take the nth rootof each side.

Solving Equations Using nth Roots

a. 2x4 = 162 b. (x º 2)3 = 10

x4 = 81 x º 2 = �3 1�0�

x = ±�4 8�1� x = �3 1�0� + 2

x = ±3 x ≈ 4.15

E X A M P L E 4

E X A M P L E 3

1�(321/5)2

1�(�5 3�2�)2

E X A M P L E 2

Let a1/n be an nth root of a, and let m be a positive integer.

• am/n = (a1/n)m = (�n a�)m

• aºm/n = = = , a ≠ 01�(�n a�)m

1�(a1/n)m

1�am/n

RATIONAL EXPONENTS

STUDENT HELP

Study Tip

To use a scientificcalculator in Example 3,replace with

and replacewith .

3.343701525



USING N TH ROOTS IN REAL LIFE

Evaluating a Model with nth Roots

The total mass M (in kilograms) of aspacecraft that can be propelled by amagnetic sail is, in theory, given by

M =

where m is the mass (in kilograms) of themagnetic sail, f is the drag force (in newtons)of the spacecraft, and d is the distance (inastronomical units) to the sun. Find the totalmass of a spacecraft that can be sent to Marsusing m = 5000 kg, f = 4.52 N, and d = 1.52 AU. � Source: Journal of Spacecraft and Rockets

SOLUTION

M = Write model for total mass.

= Substitute for m, f, and d.

≈ 47,500 Use a calculator.

� The spacecraft can have a total mass of about 47,500 kilograms. (For comparison, the liftoff weight for a space shuttle is usually about 2,040,000 kilograms.)

Solving an Equation Using an nth Root

NAUTICAL SCIENCE The Olympias is a reconstruction of a trireme, a type ofGreek galley ship used over 2000 years ago. The power P (in kilowatts) needed topropel the Olympias at a desired speed s (in knots) can be modeled by this equation:

P = 0.0289s3

A volunteer crew of the Olympias was able to generate a maximum power of about10.5 kilowatts. What was their greatest speed? � Source: Scientific American

SOLUTION

P = 0.0289s3 Write model for power.

10.5 = 0.0289s3 Substitute 10.5 for P.

363 ≈ s3 Divide each side by 0.0289.

�3 3�6�3� ≈ s Take cube root of each side.

7 ≈ s Use a calculator.

� The greatest speed attained by the Olympias was approximately 7 knots (about 8 miles per hour).

E X A M P L E 6

0.015(5000)2��4.52(1.52)4/3

0.015m2�

fd4/3

0.015m2�

fd 4/3

E X A M P L E 5

GOAL 2

RE

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RE

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Space Science

NAUTICALSCIENCE The

Olympias was completedand first launched in 1987. A crew of 170 rowers isneeded to run the ship.

APPLICATION LINKwww.mcdougallittell.com

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FOCUS ONAPPLICATIONS

Artist’s rendition of a magnetic sail



1. What is the index of a radical?

2. LOGICAL REASONING Let n be an integer greater than 1. Tell whether thegiven statement is always true, sometimes true, or never true. Explain.

a. If xn = a, then x = �n a�. b. a1/n = �a1n�

3. Try to evaluate the expressions º�46�2�5� and �4

º�6�2�5�. Explain the difference inyour results.

Evaluate the expression.

4. �4 8�1� 5. º(491/2) 6. (�3 º�8�)5 7. 31252/5

Solve the equation.

8. x3 = 125 9. 3x5 = º3 10. (x + 4)2 = 0 11. x4 º 7 = 9993

12. SHOT PUT The shot (a metal sphere) used in men’s shot put has a volumeof about 905 cubic centimeters. Find the radius of the shot. (Hint: Use the formula V = �

43�πr3 for the volume of a sphere.)

USING RATIONAL EXPONENT NOTATION Rewrite the expression usingrational exponent notation.

13. �41�4� 14. �3

1�1� 15. (�75�)2 16. (�9

1�6�)5 17. (�82�)11

USING RADICAL NOTATION Rewrite the expression using radical notation.

18. 61/3 19. 71/4 20. 103/7 21. 52/5 22. 87/4

FINDING NTH ROOTS Find the indicated real nth root(s) of a.

23. n = 2, a = 100 24. n = 4, a = 0 25. n = 3, a = º8

26. n = 7, a = 128 27. n = 6, a = º1 28. n = 5, a = 0

EVALUATING EXPRESSIONS Evaluate the expression without using acalculator.

29. �36�4� 30. �3

º�1�0�0�0� 31. º�66�4�

32. 4º1/2 33. 11/3 34. º(2561/4)35. (�4

1�6�)2 36. (�3º�2�7�)º4 37. (�6

0�)3

38. º(25º3/2) 39. 324/5 40. (º125)º2/3

APPROXIMATING ROOTS Evaluate the expression using a calculator.Round the result to two decimal places when appropriate.

41. �5º�1�6�,8�0�7� 42. �9

1�1�2�4� 43. �86�5�,5�3�6�

44. 41/10 45. 10º1/4 46. º(13311/3)47. (�3

1�1�2�)º4 48. ( 7�º�2�8�0�)3 49. (�66�)2

50. (º190)º4/5 51. 26º3/4 52. 5222/7

PRACTICE AND APPLICATIONS

GUIDED PRACTICE

Vocabulary Check ✓Concept Check ✓

Skill Check ✓

Extra Practice to help you masterskills is on p. 949.

STUDENT HELP

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 13–28Example 2: Exs. 29–40Example 3: Exs. 41–52Example 4: Exs. 53–61Example 5: Exs. 62–64Example 6: Exs. 65–67



SOLVING EQUATIONS Solve the equation. Round your answer to two decimalplaces when appropriate.

53. x5 = 243 54. 6x3 = º1296 55. x6 + 10 = 10

56. (x º 4)4 = 81 57. ºx7 = 40 58. º12x4 = º48

59. (x + 12)3 = 21 60. x3 º 14 = 22 61. x8 º 25 = º10

62. For mammals, the lung volume V (in milliliters) can be modeled by V = 170m4/5 where m is the body mass (in kilograms). Find the lung volume of eachmammal in the table shown. � Source: Respiration Physiology

63. SPILLWAY OF A DAM A dam’s spillway capacity is an indication of howthe dam will perform under certain flood conditions. The spillway capacity q (incubic feet per second) of a dam can be calculated using the formula q = c¬h3/2

where c is the discharge coefficient, ¬ is the length (in feet) of the spillway, and h is the height (in feet) of the water on the spillway. A dam with a spillway 40 feet long, 5 feet deep, and 5 feet wide has a discharge coefficient of 2.79. What is the dam’s maximum spillway capacity? � Source: Standard Handbook for Civil Engineers

64. INFLATION If the price of an item increases from p1 to p2 over a period of n years, the annual rate of inflation i (expressed as a decimal) can be modeled

by i = � �1/nº 1. In 1940 the average value of a home was $2900. In 1990 the

average value was $79,100. What was the rate of inflation for a home?� Source: Bureau of the Census

65. The formula for the volume Vof a regular dodecahedron (a solid with 12 regularpentagons as faces) is V ≈ 7.66a3 where a is the length ofan edge of the dodecahedron. Find the length of an edge ofa regular dodecahedron that has a volume of 30 cubic feet.Round your answer to two decimal places.

66. The formula for the volume Vof a regular icosahedron (a solid with 20 congruentequilateral triangles as faces) is V ≈ 2.18a3 where a is thelength of an edge of the icosahedron. Find the length of anedge of a regular icosahedron that has a volume of 21 cubiccentimeters. Round your answer to two decimal places.

67. ISLAND SPECIES Philip Darlington discovered a rule of thumb that relates an island’s land area A (in square miles) to the number s of reptile and amphibian species the island can support by the model A = 0.0779s3. The area of Puerto Rico is roughly 4000 square miles. About how many reptile and amphibian species can it support? � Source: The Song of the Dodo: Island Biogeography in an Age of Extinctions

GEOMETRY CONNECTION

GEOMETRY CONNECTION

p2�p1

BIOLOGY CONNECTION

h

¬

HOMEWORK HELPVisit our Web site

www.mcdougallittell.comfor help with problemsolving in Ex. 64.

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STUDENT HELP

Mammal Body mass (kg)

Banded mongoose 1.14

Camel 229

Horse 510

Swiss cow 700


68. MULTI-STEP PROBLEM A board foot is a unit for measuring wood. One board foot has a volume of 144 cubic inches. The Doyle log rule,

given by b = l��r º

22

��2, is a formula for

approximating the number b of board feet in a log with length l (in feet) and radius r (in inches). The total volume V (in cubic inches) of wood in the main trunk of a Douglas fir can be modeled by V = 250r3 where r is the radiusof the trunk at the base of the tree. Suppose you need 5000 board feet from a20 foot Douglas fir log.

a. What volume of wood do you need?

b. What is the radius of a log that will meet your needs?

c. What is the total volume of wood in the main trunk of a Douglas fir tree thatwill meet your needs?

d. If you find a suitable tree, what fraction of the tree would you actually use?

e. Writing How does your answer to part (d) change if you instead needonly 2500 board feet?

69. VISUAL THINKING Copy the table. Give thenumber of nth roots of a for each category.

70. The graph of y = xn where n is even is shown in red.Explain how the graph justifies the table for n even.

71. Draw a similar graph to justify the table for n odd.

SOLVING SYSTEMS Use Cramer’s rule to solve the linear system. (Review 4.3)

72. x + 4y = 12 73. x º 2y = 11 74. 2x º 4y = 72x + 5y = 18 2x + 5y = º14 ºx + y = 1

75. º3x + 2y = º9 76. ºx º 8y = 10 77. ºx º y = 0x º 4y = 2 10x + y = 1 5x º 6y = 13

SIMPLIFYING EXPRESSIONS Simplify the expression. Tell which properties ofexponents you used. (Review 6.1 for 7.2)

78. x4 • xº2 79. (xº3)5 80. (2xy3)º2 81. 5xº2y0

82. �xxº

3

4� 83. ��xº

y

2��2

84. 85. •

FINDING ZEROS Find all the zeros of the polynomial function. (Review 6.7)

86. ƒ(x) = x4 + 9x3 º 5x2 º 153x º 140 87. ƒ(x) = x4 + x3 º 19x2 + 11x + 30

88. ƒ(x) = x3 º 5x2 + 16x º 80 89. ƒ(x) = x3 º x2 + 9x º 9

9x6y�

4y16xy�9x5

7x3y8�14xyº2

MIXED REVIEW


y

x

y � aa > 0

y � 0

y � aa < 0

TestPreparation

★★ Challenge

EXTRA CHALLENGE

www.mcdougallittell.com

Log-sawing patterns formaximum board feet

Ex. 70

a < 0 a = 0 a > 0

n is even ? ? ?

n is odd ? ? ?


Properties of RationalExponents

PROPERTIES OF RATIONAL EXPONENTS AND RADICALS

The properties of integer exponents presented in Lesson 6.1 can also be applied torational exponents.

If m = �1n� for some integer n greater than 1, the third and sixth properties can be

written using radical notation as follows:

�na� •� b� = �n

a� • �nb� Product property

�n � = Quotient property

Using Properties of Rational Exponents

Use the properties of rational exponents to simplify the expression.

a. 51/2 • 51/4 = 5(1/2 + 1/4) = 53/4

b. (81/2 • 51/3)2 = (81/2)2 • (51/3)2 = 8(1/2 • 2) • 5(1/3 • 2) = 81 • 52/3 = 8 • 52/3

c. (24 • 34)º1/4 = �(2 • 3)4�º1/4 = (64)º1/4 = 6[4 • (º1/4)] = 6º1 = �61

�

d. �771/3� = �

771

1

/3� = 7(1 º 1/3) = 72/3

e. � 2 = ��

142�1/3�2

= (31/3)2 = 3(1/3 • 2) = 32/3121/3�41/3

E X A M P L E 1

�na�

��n

b�a�b

GOAL 1

7.2 Properties of Rational Exponents 407

Use properties ofrational exponents toevaluate and simplifyexpressions.

Use properties ofrational exponents to solvereal-life problems, such asfinding the surface area of amammal in Example 8.

� To model real-lifequantities, such as thefrequencies in the musicalrange of a trumpet for Ex. 94.


GOAL 2

GOAL 1


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Let a and b be real numbers and let m and n be rational numbers. The followingproperties have the same names as those listed on page 323, but now apply torational exponents as illustrated.

PROPERTY EXAMPLE

1. am • an = am + n 31/2 • 33/2 = 3(1/2 + 3/2) = 32 = 9

2. (am)n = amn (43/2)2 = 4(3/2 • 2) = 43 = 64

3. (ab)m = ambm (9 • 4)1/2 = 91/2 • 41/2 = 3 • 2 = 6

4. aºm = , a ≠ 0 25º1/2 = = �51

�

5. = am º n, a ≠ 0 �66

5

1

/

/

2

2� = 6(5/2 º 1/2) = 62 = 36

6. ��ba

�m= , b ≠ 0 ��2

87�1/3

= �287

1

1

/3

/3� = �32

�am�bm

am�an

1�251/2

1�am

PROPERTIES OF RATIONAL EXPONENTSCONCEPT

SUMMARY

Look Back For help with propertiesof exponents, see p. 324.

STUDENT HELP



Using Properties of Radicals

Use the properties of radicals to simplify the expression.

a. �3 4� • �3 1�6� = �3 4� •� 1�6� = �3 6�4� = 4 Use the product property.

b. = �4 � = �4 8�1� = 3 Use the quotient property.

. . . . . . . . . .

For a radical to be in you must not only apply the properties ofradicals, but also remove any perfect nth powers (other than 1) and rationalize anydenominators.

Writing Radicals in Simplest Form

Write the expression in simplest form.

a. �3 5�4� = �3 2�7� •� 2� Factor out perfect cube.

= �3 2�7� • �3 2� Product property

= 3�3 2� Simplify.

b. �5 � = �5 � Make the denominator a perfect fifth power.

= �5 � Simplify.

= Quotient property

= Simplify.

. . . . . . . . . .

Two radical expressions are if they have the same index and the same

radicand. For instance, �3 2� and 4�3 2� are like radicals. To add or subtract likeradicals, use the distributive property.

Adding and Subtracting Roots and Radicals

Perform the indicated operation.

a. 7(61/5) + 2(61/5) = (7 + 2)(61/5) = 9(61/5)b. �3 1�6� º �3 2� = �3 8� •� 2� º �3 2�

= �3 8� • �3 2� º �3 2�

= 2�3 2� º �3 2�

= (2 º 1)�3 2�

= �3 2�

E X A M P L E 4

like radicals

�52�4�

�2

�5 2�4��5 3�2�

24�32

3 • 8�4 • 8

3�4

E X A M P L E 3

simplest form,

162�2

�4 1�6�2��

�4 2�

E X A M P L E 2


www.mcdougallittell.comfor extra examples.

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The properties of rational exponents and radicals can also be applied to expressionsinvolving variables. Because a variable can be positive, negative or zero, sometimesabsolute value is needed when simplifying a variable expression.

�nx�n� = x when n is odd �7 2�7� = 2 and �7 (º�2�)7� = º2

�nx�n� = |x| when n is even �4 5�4� = 5 and �4 (º�5�)4� = 5

Absolute value is not needed when all variables are assumed to be positive.

Simplifying Expressions Involving Variables

Simplify the expression. Assume all variables are positive.

a. �3 1�2�5�y6� = �3 53�(y�2)�3� = 5y2

b. (9u2v10)1/2 = 91/2 (u2)1/2(v10)1/2 = 3u(2 • 1/2)v(10 • 1/2) = 3uv5

c. �4 � = = =

d. �2

6

x1

x/

y3

1

z

/

º

2

5� = 3x(1 º 1/3)y1/2zº(º5) = 3x2/3y1/2z5

Writing Variable Expressions in Simplest Form

Write the expression in simplest form. Assume all variables are positive.

a. �5 5�a�5b�9c�13� = �5 5�a�5b�5b�4c�10�c3� Factor out perfect fifth powers.

= �5 a�5b�5c�10� • �5 5�b�4c�3� Product property

= abc2�5 5�b�4c�3� Simplify.

b. �3 � = �3 � Make the denominator a perfect cube.

= �3 � Simplify.

= Quotient property

= Simplify.

Adding and Subtracting Expressions Involving Variables

Perform the indicated operation. Assume all variables are positive.

a. 5�y� + 6�y� = (5 + 6)�y� = 11�y�

b. 2xy1/3 º 7xy1/3 = (2 º 7)xy1/3 = º5xy1/3

c. 3�3 5�x5� º x�3 4�0�x2� = 3x�3 5�x2� º 2x�3 5�x2� = (3x º 2x)�3 5�x2� = x�3 5�x2�

E X A M P L E 7

�3 xy�2��

y3

�3 xy�2��3 y9�

xy2�y9

xy2�y7y2

x�y7

E X A M P L E 6

x�y2

�4 x4��4 (y�2)�4�

�4 x4��4 y8�

x4�y8

E X A M P L E 5



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PROPERTIES OF RATIONAL EXPONENTS IN REAL LIFE

Evaluating a Model Using Properties of Rational Exponents

Biologists study characteristics of various living things. One way of comparingdifferent animals is to compare their sizes. For example, a mammal’s surface area S(in square centimeters) can be approximated by the model S = km2/3 where m is themass (in grams) of the mammal and k is a constant. The values of k for severalmammals are given in the table.

Approximate the surface area of a cat that has a mass of 5 kilograms(5 ª 103 grams). � Source: Scaling: Why Is Animal Size So Important?

SOLUTION

S = km2/3 Write model.

= 10.0(5 ª 103)2/3 Substitute for k and m.

= 10.0(5)2/3(103)2/3 Power of a product property

≈ 10.0(2.92)(102) Power of a power property

= 2920 Simplify.

� The cat’s surface area is approximately 3000 square centimeters.

Using Properties of Rational Exponents with Variables

BIOLOGY CONNECTION You are studying a Canadian lynx whose mass is twice themass of an average house cat. Is its surface area also twice that of an average house cat?

SOLUTION

Let m be the mass of an average house cat. Then the mass of the Canadian lynx is 2m.The surface areas of the house cat and the Canadian lynx can be approximated by:

Scat = 10.0m2/3 Slynx = 10.0(2m)2/3

To compare the surface areas look at their ratio.

= Write ratio of surface areas.

= Power of a product property

= 22/3 Simplify.

≈ 1.59 Evaluate.

� The surface area of the Canadian lynx is about one and a half times that of anaverage house cat, not twice as much.

10.0(22/3)(m2/3)��

10.0m2/3

10.0(2m)2/3��

10.0m2/3

Slynx�Scat

E X A M P L E 9

E X A M P L E 8

GOAL 2

REAL LIFE

REAL LIFE

Biology

BIOLOGY Theaverage mass of a

Canadian lynx is about9.1 kilograms. The averagemass of a house cat is about4.8 kilograms.

RE

AL LIFE

RE

AL LIFE


Mammal Mouse Cat Large dog Cow Rabbit Human

k 9.0 10.0 11.2 9.0 9.75 11.0



1. List three pairs of like radicals.

2. If you know that 46,656,000 = 29 • 36 • 53, what is the cube root of 46,656,000?Explain your reasoning.

ERROR ANALYSIS Explain the error made in simplifying the expression.

3. 3�4 5� + 2�4 5� = 5�4 10� 4. � 1/3 = =

Simplify the expression.

5. 31/4 • 33/4 6. (51/3)6 7. �3 1�6� • �3 4� 8. 4º1/2

9. �4 � 10. �3 � 11. 81/7 + 2(81/7) 12. �2�0�0� º 3�2�


13. x2/3 • x4/3 14. (y1/6)3 15. �4�a�6� 16. bº1/3

17. �5 � 18. �3 � 19. 2a1/5 º 6a1/5 20. x�3 y6� + y2�3 x3�

21. The average mass of a rabbit is 1.6 kilograms. Use theinformation given in Example 8 to approximate the surface area of a rabbit.

PROPERTIES OF RATIONAL EXPONENTS Simplify the expression.

22. 35/3 • 31/3 23. (52/3)1/2 24. 41/4 • 641/4 25.

26. 27. 28. (21/4 • 21/3)6 29. � º1/2

30. 31. 32. 33. (103/4 • 43/4)º4

PROPERTIES OF RADICALS Simplify the expression.

34. �6�4� • �3 6�4� 35. �4 8� • �4 2� 36. �4 5� • �4 5� 37. (�3 6� • �4 6�)12

38. 39. 40. 41.

SIMPLEST FORM Write the expression in simplest form.

42. �5�0� 43. �5 1�2�1�5� 44. �3 1�8� • �3 1�5� 45. 3�4 2�4� • 5�4 2�

46. �3 � 47. 48. 4�� 49.

COMBINING ROOTS AND RADICALS Perform the indicated operation.

50. �5 6� + 5�5 6� 51. 5(5)1/7 º 7(5)1/7 52. º�8 4� + 5�8 4�

53. 1601/2 º 101/2 54. �3 3�7�5� + �3 8�1� 55. 2�4 1�7�6� + 5�4 1�1�

�3 4��5 8�

80�9

2��6 8�1�

1�7

�3 9� • �3 6��6 2� • �6 2�

�6 8� • �6 1�6��

�6 2��3 4��3 3�2�

�7��5 7�

1210/8�12º3/8

1252/9 • 1251/9��

51/462/3 • 42/3�

32/3

52�82

701/3�141/3

71/5�73/5

1�36º1/2


BIOLOGY CONNECTION

x2�z

x10�y5

1�4

16�81

x1/3�y2

x 1/3

�(y8)1/3x

�y8

GUIDED PRACTICE


Skill Check ✓

STUDENT HELP

Extra Practice to help you masterskills is on p. 949.

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 22– 33Example 2: Exs. 34– 41Example 3: Exs. 42– 49Example 4: Exs. 50–55Example 5: Exs. 56– 67Example 6: Exs. 68–75Example 7: Exs. 76–81Example 8: Exs. 90–93Example 9: Exs. 94– 97



VARIABLE EXPRESSIONS Simplify the expression. Assume all variables arepositive.

56. x1/3 • x1/5 57. (y3)1/6 58. �5 3�2�x5� 59.

60. 61. 4��x

y�1

4

2�� 62. 63. (y • y1/4)4/3

64. (�4 x3� • �4 x5�)º265. 66. 67.

SIMPLEST FORM Write the expression in simplest form. Assume all variablesare positive.

68. �3�6�x3� 69. �4 1�0�x5�y8�z1�0� 70. �5 8�xy�7� • �5 6�x6� 71. �xy�z� • �2�y3�z4�

72. 73. �3 � 74. �� 75.

COMBINING VARIABLE EXPRESSIONS Perform the indicated operation.Assume all variables are positive.

76. 2�5 y� + 7�5 y� 77. 9x1/5 º 2x1/5 78. º�4 x� + 2�4 x�

79. (x9y)1/3 + (xy1/9)3 80. �4�x5� º x�x3� 81. y�3 2�4�x5� + �3 º�3�x2�y3�

EXTENSION: IRRATIONAL EXPONENTS The properties you studied in thislesson can also be applied to irrational exponents. Simplify the expression.Assume all variables are positive.

82. x2 • x�3� 83. (y�2�)�2� 84. (xy)π 85. 4º�7�

86. 87. � π88. 3x�2� + x�2� 89. xy�1�1� º 3xy�1�1�

90. Find a radical expression forthe perimeter of the triangle. Simplify the expression.

91. The areas of two circles are15 square centimeters and 20 square centimeters. Findthe exact ratio of the radius of the smaller circle to theradius of the larger circle.

92. Look back at Example 8. Approximate the surface areaof a human that has a mass of 68 kilograms.

93. PINHOLE CAMERA A pinhole camera is made out of a light-tight boxwith a piece of film attached to one side and a pinhole on the opposite side.The optimum diameter d (in millimeters) of the pinhole can be modeled by d = 1.9�(5.5 ª 10º4)l�1/2, where l is the length of the camera box (in millimeters). What is the optimum diameter for a pinhole camera if the camera box has a length of 10 centimeters?

BIOLOGY CONNECTION

GEOMETRY CONNECTION

GEOMETRY CONNECTION

x1/π�y2/π

x2�5��x�5�

�5 x3��7 x4�

9x2y�32z3

x3�y2

4��3 x�

�3 y6��3 2�7�y� • �3 y1�1�

2�x� • �x3��

�9�x1�0�x3/4yzº1/3��

x1/4z2/3

x5/3y�xyº1/2

x3/7�x1/3

1�xº5/4

12 2

4

film

pinhole

tree

l

Skills Review For help with perimeterand area, see p. 914.

STUDENT HELP

Ex. 90



MUSIC In Exercises 94 and 95, use the following information.The musical note A-440 (the A above middle C) has a frequency of 440 vibrationsper second. The frequency ƒ of any note can be found using ƒ = 440 • 2n/12 where n represents the number of black and white keys the given note is above or belowA-440. For notes above A-440, n > 0, and for notes below A-440, n < 0.

94. Find the highest and lowest frequencies in the musical range of a trumpet. What is the exact ratio of these two frequencies?

95. LOGICAL REASONING Describe the pattern of the frequencies of successivenotes with the same letter.

96. DISTANCE OF AN OBJECT The maximum horizontal distance d that anobject can travel when launched at an optimum angle of projection from an

initial height h0 can be modeled by d = where v0 is the initial

speed and g is the acceleration due to gravity. Simplify the model when h0 = 0.

97. BALLOONS You have filled two round balloons with air. One balloon hastwice as much air as the other balloon. The formula for the surface area S of asphere in terms of its volume V is S = (4π)1/3(3V)2/3. By what factor is thesurface area of the larger balloon greater than that of the smaller balloon?

98. MULTI-STEP PROBLEM A common ant absorbsoxygen at a rate of about 6.2 milliliters per second per square centimeter of exoskeleton. It needs about24 milliliters of oxygen per second per cubiccentimeter of its body. An ant is basically cylindricalin shape, so its surface area S and volume V can beapproximated by the formulas for the surface area andvolume of a cylinder:

S = 2πrh + 2πr2 V = πr2h

a. Approximate the surface area and volume of an ant that is 8 millimeters longand has a radius of 1.5 millimeters. Would this ant have a surface area largeenough to meet its oxygen needs?

b. Consider a “giant” ant that is 8 meters long and has a radius of 1.5 meters.Would this ant have a surface area large enough to meet its oxygen needs?

c. Writing Substitute 1000r for r and 1000h for h into the formulas for surface area and volume. How does increasing the radius and height by a factor of 1000 affect surface area? How does it affect volume? Use theresults to explain why “giant” ants do not exist.

99. CRITICAL THINKING Substitute different combinations of odd and even positive integers for m and n in the expression �n

xm�. Do not assume that x isalways positive. When is absolute value needed in simplifying the expression?

v0�(v�0)�2�+� 2�g�h�0��g

�48

mid

dle

C

A-44

0

trumpet

�36 �24 �12 0 12 24 36

A B C D E F G A B B B B B B BC C C C C C CD D D D D DE E E E E EF G A F G A F G A F G A F G A F G A

TestPreparation

★★ Challenge



COMPLETING THE SQUARE Find the value of c that makes the expression aperfect square trinomial. Then write the expression as the square of a binomial.(Review 5.5)

100. x2 + 14x + c 101. x2 º 21x + c 102. x2 º 7.6x + c

103. x2 + 9.9x + c 104. x2 + �23�x + c 105. x2 º �

14�x + c

POLYNOMIAL OPERATIONS Perform the indicated operation. (Review 6.3 for 7.3)

106. (º3x3 + 6x) º (8x3 + x2 º 4x) 107. (50x º 3) + (8x3 + 9x2 + 2x + 4)

108. 20x2(x º 9) 109. (2x + 7)2

LONG DIVISION Divide using long division. (Review 6.5 for 7.3)

110. (x3 º 28x º 48) ÷ (x + 4) 111. (4x2 + 3x º 3) ÷ (x + 1)

112. (4x2 º 6x) ÷ (x º 2) 113. (x4 º 2x3 º 70x + 20) ÷ (x º 5)

Evaluate the expression without using a calculator. (Lesson 7.1)

1. 82/3 2. 32º3/5 3. º(811/4) 4. (º64)2/3

Solve the equation. Round your answer to two decimal places. (Lesson 7.1)

5. x5 = 10 6. º9x6 = –18 7. x4 º 4 = 9 8. (x + 2)3 = º15

Write the expression in simplest form. (Lesson 7.2)

9. 10. �4 � 11.

12. �4�5� 13. �3 7� • �3 4�9� 14. 81/5 + 2(81/5)Write the expression in simplest form. Assume all variables are positive. (Lesson 7.2)

15. �3 x2� • �4 x� 16. (x1/5)5/2 17.

18. �3 5�x3�y5� 19. �� 20. x(9y)1/2 º (x2y)1/2

21. GENERATING POWER As a rule of thumb, the power P (in horsepower)

that a ship needs can be modeled by P = where d is the ship’s

displacement (in tons), s is the normal speed (in knots), and c is the Admiraltycoefficient. If a ship displaces 30,090 tons, has a normal speed of 22.5 knots, and has an Admiralty coefficient of 370, how much power does it need? (Lesson 7.1)

22. The surface area S (in square centimeters) of a large dogcan be approximated by the model S = 11.2m2/3 where m is the mass (in grams)of the dog. A Labrador retriever’s mass is about three times the mass of aScottish terrier. Is its surface area also three times that of a Scottish terrier?(Lesson 7.2)

BIOLOGY CONNECTION

d2/3 • s3�c

36x�y3

xy1/2�x3/4yº2

5121/3�

81/316�3

1�4º1/4

QUIZ 1 Self-Test for Lessons 7.1 and 7.2

MIXED REVIEW


7.3 Power Functions and Function Operations 415

Power Functions and FunctionOperations

PERFORMING FUNCTION OPERATIONS

In Chapter 6 you learned how to add, subtract, multiply, and divide polynomialfunctions. These operations can be defined for any functions.

So far you have studied various types of functions, including linear functions,quadratic functions, and polynomial functions of higher degree. Another commontype of function is a which has the form y = axb where a is a realnumber and b is a rational number.

Note that when b is a positive integer, a power function is simply a type ofpolynomial function. For example, y = axb is a linear function when b = 1, a quadratic function when b = 2, and a cubic function when b = 3.

Adding and Subtracting Functions

Let ƒ(x) = 2x1/2 and g(x) = º6x1/2. Find (a) the sum of the functions, (b) thedifference of the functions, and (c) the domains of the sum and difference.

SOLUTION

a. ƒ(x) + g(x) = 2x1/2 + (º6x1/2) = (2 º 6)x1/2 = º4x1/2

b. ƒ(x) º g(x) = 2x1/2 º (º6x1/2) = �2 º (º6)�x1/2 = 8x1/2

c. The functions ƒ and g each have the same domain—all nonnegative realnumbers. So, the domains of ƒ + g and ƒ º g also consist of all nonnegative real numbers.

E X A M P L E 1

power function,

GOAL 1

Perform operationswith functions includingpower functions.

Use powerfunctions and functionoperations to solve real-lifeproblems, such as finding theproportion of water loss in abird’s egg in Example 4.

� To solve real-lifeproblems, such as finding the height of a dinosaur in Ex. 56.


GOAL 2

GOAL 1


7.3RE

AL LIFE

RE

AL LIFE

Let ƒ and g be any two functions. A new function h can be defined byperforming any of the four basic operations (addition, subtraction,multiplication, and division) on ƒ and g.

Operation Definition Example: ƒ(x ) = 2x, g(x ) = x + 1

ADDITION h(x) = ƒ(x) + g(x) h(x) = 2x + (x + 1) = 3x + 1

SUBTRACTION h(x) = ƒ(x) º g(x) h(x) = 2x º (x + 1) = x º 1

MULTIPLICATION h(x) = ƒ(x) • g(x) h(x) = (2x)(x + 1) = 2x2 + 2x

DIVISION h(x) = �gƒ(

(xx))� h(x) = �x

2+x

1�

The domain of h consists of the x-values that are in the domains of both ƒ and g. Additionally, the domain of a quotient does not include x-values for which g(x) = 0.

OPERATIONS ON FUNCTIONSCONCEPT

SUMMARY



Multiplying and Dividing Functions

Let ƒ(x) = 3x and g(x) = x1/4. Find (a) the product of the functions, (b) the quotientof the functions, and (c) the domains of the product and quotient.

SOLUTION

a. ƒ(x) • g(x) = (3x)(x1/4) = 3x(1 + 1/4) = 3x5/4

b. �ƒg

((xx))� = = 3x(1 º 1/4) = 3x3/4

c. The domain of ƒ consists of all real numbers and the domain of g consists of all nonnegative real numbers. So, the domain of ƒ • g consists of all

nonnegative real numbers. Because g(0) = 0, the domain of is restricted to all positive real numbers.

. . . . . . . . . .

A fifth operation that can be performed with two functions is composition.

As with subtraction and division of functions, you need to pay attention to the orderof functions when they are composed. In general, ƒ(g(x)) is not equal to g(ƒ(x)).

Finding the Composition of Functions

Let ƒ(x) = 3xº1 and g(x) = 2x º 1. Find the following.

a. ƒ(g(x)) b. g(ƒ(x)) c. ƒ(ƒ(x)) d. the domain of each composition

SOLUTION

a. ƒ(g(x)) = ƒ(2x º 1) = 3(2x º 1)º1 = �2x3º 1�

b. g(ƒ(x)) = g(3xº1) = 2(3xº1) º 1 = 6xº1 º 1 = �6x� º 1

c. ƒ(ƒ(x)) = ƒ(3xº1) = 3(3xº1)º1 = 3(3º1x) = 30x = x

d. The domain of ƒ(g(x)) consists of all real numbers except x = �12� because

g��12�� = 0 is not in the domain of ƒ. The domains of g(ƒ(x)) and ƒ(ƒ(x)) consist

of all real numbers except x = 0, because 0 is not in the domain of ƒ. Note thatƒ(ƒ(x)) simplifies to x, but that result is not what determines the domain.

E X A M P L E 3

ƒ�g

3x�x1/4

E X A M P L E 2

The of the function ƒ with the function g is:

h(x) = ƒ(g(x))

The domain of h is the set of all x-values such that x is in the domain of g and g(x) is in the domain of ƒ.

composition

COMPOSITION OF TWO FUNCTIONS

STUDENT HELP

Study TipWhen you are writing thecomposition of ƒ with g,you may want to firstrewrite ƒ(x) = 3xº1 as

ƒ( ) = 3( )º1

and then substituteg(x) = 2x º 1 every-where there is a box.

Input

domain of g

x

Output

range of g

Input

domain of ƒ range of ƒ

ƒ(g(x))g(x)

Output

Look Back For help with functionoperations, see p. 338.

STUDENT HELP


USING FUNCTION OPERATIONS IN REAL LIFE

Using Function Operations

BIOLOGY CONNECTION You are doing a science project and have found researchindicating that the incubation time I (in days) of a bird’s egg can be modeled by I(m) = 12m0.217 where m is the egg’s mass (in grams). You have also found thatduring incubation the egg’s rate of water loss R (in grams per day) can be modeled byR(m) = 0.015m0.742.

You conjecture that the proportion of water loss during incubation is about the samefor any size egg. Show how you can use the two power function models to verifyyour conjecture. � Source: Biology by Numbers

SOLUTION

= =

Daily water loss = Number of days = Egg’s mass =

=

=

= 0.18mº0.041

Because mº0.041 is approximately m0, the proportion of water loss can be treated as0.18m0 = (0.18)(1) = 0.18. So, the proportion of water loss is about 18% for anysize bird’s egg, and your conjecture is correct.

Using Composition of Functions

A clothing store advertises that it is having a 25% off sale. For one day only, the storeadvertises an additional savings of 10%.

a. Use composition of functions to find the total percent discount.

b. What would be the sale price of a $40 sweater?

SOLUTION

a. Let x represent the price. The sale price for a 25% discount can be represented bythe function ƒ(x) = x º 0.25x = 0.75x. The reduced sale price for an additional10% discount can be represented by the function g(x) = x º 0.10x = 0.90x.

g(ƒ(x)) = g(0.75x) = 0.90(0.75x) = 0.675x

� The total percent discount is 100% º 67.5% = 32.5%.

b. Let x = 40. Then g(ƒ(x)) = g(ƒ(40)) = 0.675(40) = 27.

� The sale price of the sweater is $27.

E X A M P L E 5

0.18m0.959��m

(0.015m0.742)(12m0.217)��m

mI(m)R(m)

Proportionof water

loss

E X A M P L E 4

GOAL 2


VERBALMODEL

ALGEBRAICMODEL

LABELS

RE

AL LIFE

RE

AL LIFE

Business

•��

Egg’s mass

Number of daysDaily water loss��

Egg’s mass

Total water loss

• ��

m

I(m)R(m)

BIOLOGY Thelargest bird egg

is an ostrich’s egg, which has a mass of about 1.65 kilograms. For comparison, an averagechicken egg has a mass of only 56.7 grams.

RE

AL LIFE

RE

AL LIFE

Skills Review For help with calculatingpercents, see p. 907.

STUDENT HELP




1. Complete this statement: The function y = axb is a(n) ��? function where a is a(n) ��? number and b is a(n) ��? number.

2. LOGICAL REASONING Tell whether the sum of two power functions issometimes, always, or never a power function.

ERROR ANALYSIS Let ƒ(x) = x2 + 2 and g(x) = 3x. What is wrong with thecomposition shown? Explain.

3. 4.

Let ƒ(x) = 4x and g(x) = x º 1. Perform the indicated operation and state the domain.

5. ƒ(x) + g(x) 6. ƒ(x) º g(x) 7. ƒ(x) • g(x)

8. �ƒg

((xx))� 9. ƒ(g(x)) 10. g(ƒ(x))

11. SALES BONUS You are a sales representative for a clothing manufacturer.You are paid an annual salary plus a bonus of 2% of your sales over $200,000.Consider two functions: ƒ(x) = x º 200,000 and g(x) = 0.02x. If x > $200,000,which composition, ƒ(g(x)) or g(ƒ(x)), represents your bonus? Explain.

ADDING AND SUBTRACTING FUNCTIONS Let ƒ(x) = x2 º 5x + 8 and g(x) = x2 º 4. Perform the indicated operation and state the domain.

12. ƒ(x) + g(x) 13. g(x) + ƒ(x) 14. ƒ(x) + ƒ(x) 15. g(x) + g(x)

16. ƒ(x) º g(x) 17. g(x) º ƒ(x) 18. ƒ(x) º ƒ(x) 19. g(x) º g(x)

MULTIPLYING AND DIVIDING FUNCTIONS Let ƒ(x) = 2x2/3 and g(x) = 3x1/2.Perform the indicated operation and state the domain.

20. ƒ(x) • g(x) 21. g(x) • ƒ(x) 22. ƒ(x) • ƒ(x) 23. g(x) • g(x)

24. �ƒg

((xx))� 25. ��ƒ

g((xx))

� 26. �ƒƒ

((xx))� 27. �g

g((xx))

�

COMPOSITION OF FUNCTIONS Let ƒ(x) = 4xº5 and g(x) = x3/4. Perform theindicated operation and state the domain.

28. ƒ(g(x)) 29. g(ƒ(x)) 30. ƒ(ƒ(x)) 31. g(g(x))

FUNCTION OPERATIONS Let ƒ(x) = 10x and g(x) = x + 4. Perform theindicated operation and state the domain.

32. ƒ(x) + g(x) 33. ƒ(x) º g(x) 34. ƒ(x) • g(x) 35. �gƒ((xx))

�

36. ƒ(g(x)) 37. g(ƒ(x)) 38. ƒ(ƒ(x)) 39. g(g(x))


GUIDED PRACTICE

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 12–19,

32–51Example 2: Exs. 20–27,

32–51Example 3: Exs. 28–51Example 4: Exs. 52, 53Example 5: Exs. 54–56

Extra Practiceto help you masterskills is on p. 949.

STUDENT HELP

f (g(x)) = f (x2 + 2) f (g(x)) = f (3x)

= 3(x2 + 2) = 3x2 + 2

= 3x2 + 6

Vocabulary Check ✓

Concept Check ✓

Skill Check ✓



FUNCTION OPERATIONS Perform the indicated operation and state the domain.

40. ƒ + g; ƒ(x) = x + 3, g(x) = 5x 41. ƒ + g; ƒ(x) = 3x1/2, g(x) = º2x1/2

42. ƒ º g; ƒ(x) = ºx2/3, g(x) = x2/3 43. ƒ º g; ƒ(x) = x2 º 3, g(x) = x + 5

44. ƒ • g; ƒ(x) = 7x2/5, g(x) = º2x3 45. ƒ • g; ƒ(x) = x º 4, g(x) = 4x2

46. �ƒg�; ƒ(x) = 9xº1, g(x) = x1/4 47. �

ƒg�; ƒ(x) = x2 º 5x, g(x) = x

48. ƒ(g(x)); ƒ(x) = 6xº1, g(x) = 5x º 2 49. g(ƒ(x)); ƒ(x) = x2 º 3, g(x) = x2 + 1

50. ƒ(ƒ(x)); ƒ(x) = 2x1/5 51. g(g(x)); g(x) = 9x º 2

52. HEART RATE For a mammal, the heart rate r (in beats per minute) and the life span s (in minutes) are related to body mass m (in kilograms) by theseformulas:

r(m) = 241mº0.25 s(m) = (6 ª 106)m0.2

Find the relationship between body mass and average number of heartbeats in alifetime by calculating r(m) • s(m). Explain the results. � Source: Physiology by Numbers

53. BREATHING RATE For a mammal, the volume b (in milliliters) of airbreathed in and the volume d (in milliliters) of the dead space (the portion of the lungs not filled with air) are related to body weight w (in grams) by theseformulas:

b(w) = 0.007w d(w) = 0.002w

The relationship between breathing rate r (in breaths per minute) and bodyweight is:

r(w) = �b(1w.1)wº

0.

d

73

(

4

w)�

Simplify r(w) and calculate the breathing rate for body weights of 6.5 grams,12,300 grams, and 70,000 grams. � Source: Respiration

COAT SALE In Exercises 54 and 55, use the following information.A clothing store is having a sale in which you can take $50 off the cost of any coat inthe store. The store also offers 10% off your entire purchase if you open a chargeaccount. You decide to open a charge account and buy a coat.

54. Use composition of functions to find the sale price of a $175 coat when $50 issubtracted before the 10% discount is applied.

55. CRITICAL THINKING Why doesn’t the store apply the 10% discount beforesubtracting $50?

56. PALEONTOLOGY The height at the hip h (in centimeters) of anornithomimid, a type of dinosaur, can be modeled by

h(l) = 3.49l1.02

where l is the length (in centimeters) of the dinosaur’s instep. The length of theinstep can be modeled by

l( f) = 1.5f

where f is the footprint length (in centimeters). Use composition of functions tofind the relationship between height and footprint length. Then find the height of an ornithomimid with a footprint length of 30 centimeters. � Source: Dinosaur Tracks

PALEONTOLOGISTA paleontologist is a

scientist who studies fossilsof dinosaurs and otherprehistoric life forms. Mostpaleontologists work ascollege professors.

CAREER LINKwww.mcdougallittell.com

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57. Writing Explain how to perform the function operations ƒ(x) + g(x),

ƒ(x) º g(x), ƒ(x) • g(x), �ƒg

((xx))�, and ƒ(g(x)) for any two functions ƒ and g.

QUANTITATIVE COMPARISON In Exercise 58–61, choose the statement that istrue about the given quantities.

¡A The quantity in column A is greater.

¡B The quantity in column B is greater.

¡C The two quantities are equal.

¡D The relationship cannot be determined from the given information.

58.

59.

60.

61.

FUNCTION COMPOSITION Find functions ƒ and g such that ƒ(g(x)) = h(x).

62. h(x) = (6x º 5)3 63. h(x) = �3x�+� 2�

64. h(x) = 65. h(x) = 3x2 + 7

66. h(x) = |2x + 9| 67. h(x) = 21x

REWRITING EQUATIONS Solve the equation for y. (Review 1.4 for 7.4)

68. y º 3x = 10 69. 2x + 3y = º8 70. x = º2y + 6

71. xy + 2 = 7 72. �12�x º �

23�y = 1 73. ax + by = c

GRAPHING FUNCTIONS Graph the function. (Review 2.1 for 7.4)

74. y = x º 2 75. y = 4x º 3 76. y = 5x º �23�

77. y = º2x º 4 78. y = º�12�x + 7 79. y = º8

SOLVING EQUATIONS Find the real-number solutions of the equation. (Review 6.4)

80. 3x3 º 2x2 = 0 81. 2x3 º 6x2 + x = 3

82. 5x4 + 19x2 º 4 = 0 83. x4 + 6x3 + 8x + 48 = 0

84. CRYPTOGRAPHY Use the inverse of A = � and the code on page 225 to decode the message. (Review 4.4)

45, 21, 84, 35, 92, 37, 142, 61

62, 25, 118, 49, 103, 44, 95, 38

21

52

MIXED REVIEW

�4 x��2

Column A Column B

ƒ(g(4)); ƒ(x) = 6x, g(x) = 3x2 ƒ(g(2)); ƒ(x) = x2/3, g(x) = º2x

g(ƒ(º1)); ƒ(x) = 5xº2, g(x) = x g(ƒ(0)); ƒ(x) = 2x + 5, g(x) = x2

ƒ(ƒ(3)); ƒ(x) = 3x º 7 ƒ(ƒ(º2)); ƒ(x) = 10x3

g(g(5)); g(x) = 16xº1/4 g(g(7)); g(x) = x2 + 8

EXTRA CHALLENGE


TestPreparation

★★ Challenge


Developing Concepts

ACTIVITY 7.4 Group Activity for use with Lesson 7.4

SET UPWork in a group of three.

MATERIALS• graph paper• straightedge

7.4 Concept Activity 421

Exploring Inverse Functions

� QUESTION How are a function and its inverse related?

� EXPLORING THE CONCEPTUse the following steps to find the inverse of ƒ(x) = �

x º2

3�.

Choose values of x and find the corresponding values of y = ƒ(x). Plot the points anddraw the line that passes through them.

Interchange the x- and y-coordinates of the ordered pairs found in Step 1. Plot thenew points and draw the line that passes through them.

Write an equation of the line from Step 2. Call this function g.

Fold your graph paper so that the graphs of ƒ and g coincide. How are the graphsgeometrically related?

In words, ƒ is the function that subtracts 3 from x and then divides the result by 2.Describe the function g in words.

Predict what the compositions ƒ(g(x)) and g(ƒ(x)) will be. Confirm your predictionsby finding ƒ(g(x)) and g(ƒ(x)).

The functions ƒ and g are inverses of each other.

� DRAWING CONCLUSIONS

Each member in your group should choose a different function from the list below.

ƒ(x) = 2x + 5 ƒ(x) = �x º

42

� ƒ(x) = 5 º �52�x

1. Complete Steps 1–3 above to find the inverse of your function.

2. Complete Step 4. How can you graph the inverse of a function without firstfinding ordered pairs (x, y)?

3. Complete Steps 5 and 6. How can you test to see if the function you found inExercise 1 is indeed the inverse of the original function?

6

5

4

3

2

1



Find inverses oflinear functions.

Find inverses ofnonlinear functions, asapplied in Example 6.

� To solve real-lifeproblems, such as findingyour bowling average in Ex. 59.


GOAL 2

GOAL 1


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Inverse FunctionsFINDING INVERSES OF LINEAR FUNCTIONS

In Lesson 2.1 you learned that a relation is a mapping of input values onto outputvalues. An maps the output values back to their original inputvalues. This means that the domain of the inverse relation is the range of the originalrelation and that the range of the inverse relation is the domain of the original relation.

Original relation Inverse relation

The graph of an inverse relation is the reflection of the graph of the original relation. The line of reflection is y = x.

Graph of original relation Reflection in y = x Graph of inverse relation

To find the inverse of a relation that is given by an equation in x and y, switch theroles of x and y and solve for y (if possible).

Finding an Inverse Relation

Find an equation for the inverse of the relation y = 2x º 4.

SOLUTION

y = 2x º 4 Write original relation.

x = 2y º 4 Switch x and y.

x + 4 = 2y Add 4 to each side.

�12

�x + 2 = y Divide each side by 2.

� The inverse relation is y = �12

�x + 2.

. . . . . . . . . .

In Example 1 both the original relation and the inverse relation happen to befunctions. In such cases the two functions are called inverse functions.

E X A M P L E 1

inverse relation

GOAL 1

y

1

1 x

y

x

y

1

1 x

x º2 º1 0 1 2

y 4 2 0 º2 º4

7.4

Look Back For help with solvingequations for y, see p. 26.

STUDENT HELP

x 4 2 0 º2 º4

y º2 º1 0 1 2


7.4 Inverse Functions 423

Given any function, you can always find its inverse relation by switching x and y. Fora linear function ƒ(x) = mx + b where m ≠ 0, the inverse is itself a linear function.

Verifying Inverse Functions

Verify that ƒ(x) = 2x º 4 and ƒº1(x) = �12

�x + 2 are inverses.

SOLUTION Show that ƒ(ƒº1(x)) = x and ƒº1(ƒ(x)) = x.

ƒ(ƒº1(x)) = ƒ��12

�x + 2� ƒº1(ƒ(x)) = ƒº1(2x º 4)

= 2��12

�x + 2� º 4 = �12

�(2x º 4) + 2

= x + 4 º 4 = x º 2 + 2

= x ✓ = x ✓

Writing an Inverse Model

When calibrating a spring scale, you need to know how farthe spring stretches based on given weights. Hooke’s lawstates that the length a spring stretches is proportional tothe weight attached to the spring. A model for one scale is ¬ = 0.5w + 3 where ¬ is the total length (in inches) of the spring and w is the weight (in pounds) of the object.

a. Find the inverse model for the scale.

b. If you place a melon on the scale and the springstretches to a total length of 5.5 inches, how much does the melon weigh?

SOLUTION

a. ¬ = 0.5w + 3 Write original model.

¬ º 3 = 0.5w Subtract 3 from each side.

�¬

0º.5

3� = w Divide each side by 0.5.

2¬ º 6 = w Simplify.

b. To find the weight of the melon, substitute 5.5 for ¬.

w = 2¬ º 6 = 2(5.5) º 6 = 11 º 6 = 5

� The melon weighs 5 pounds.

E X A M P L E 3

E X A M P L E 2

STUDENT HELP

Study TipThe notation for aninverse function, ƒº1,looks like a negativeexponent, but it shouldnot be interpreted thatway. In other words,

ƒº1(x) ≠ ( ƒ(x))º1 = .1�ƒ(x)

Functions ƒ and g are inverses of each other provided:

ƒ(g(x)) = x and g(ƒ(x)) = x

The function g is denoted by ƒº1, read as “ƒ inverse.”

INVERSE FUNCTIONS

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Study TipNotice that you do notswitch the variableswhen you are findinginverses for models. Thiswould be confusingbecause the letters arechosen to remind you ofthe real-life quantitiesthey represent.

unweightedspring

spring withweight

attached

3

0.5w

l

Not drawn to scale


FINDING INVERSES OF NONLINEAR FUNCTIONS

The graphs of the power functions ƒ(x) = x2 and g(x) = x3 are shown below alongwith their reflections in the line y = x. Notice that the inverse of g(x) = x3 is afunction, but that the inverse of ƒ(x) = x2 is not a function.

If the domain of ƒ(x) = x2 is restricted, say to only nonnegative real numbers, thenthe inverse of ƒ is a function.

Finding an Inverse Power Function

Find the inverse of the function ƒ(x) = x2, x ≥ 0.

SOLUTION

ƒ(x) = x2 Write original function.

y = x2 Replace ƒ(x) with y.

x = y2 Switch x and y.

��x = y Take square roots of each side.

Because the domain of ƒ is restricted to nonnegative values, the inverse function is ƒº1(x) = �x� . (You would choose ƒº1(x) = º�x� if the domain had been restricted to x ≤ 0.)

✓ CHECK To check your work, graph ƒ and ƒº1 as shown.

Note that the graph of ƒº1(x) = �x� is the reflection ofthe graph of ƒ(x) = x2, x ≥ 0 in the line y = x.

. . . . . . . . . .

In the graphs at the top of the page, notice that the graph of ƒ(x) = x2 can beintersected twice with a horizontal line and that its inverse is not a function. On theother hand, the graph of g(x) = x3 cannot be intersected twice with a horizontal lineand its inverse is a function. This observation suggests the horizontal line test.

E X A M P L E 4

1

1

x

y

g (x) � x 3

g�1(x) � �x3

2

2

x

y

ƒ (x) � x 2

x � y 2

GOAL 2


If no horizontal line intersects the graph of a function ƒ more than once, then theinverse of ƒ is itself a function.

HORIZONTAL LINE TEST

ƒ (x) � x2

x ≥ 0

1

1

y

ƒ�1(x) � �x

x

Look Back For help with recognizingwhen a relationship is afunction, see p. 70.

STUDENT HELP



Finding an Inverse Function

Consider the function ƒ(x) = �12

�x3 º 2. Determine whether the inverse of ƒ is a

function. Then find the inverse.

SOLUTION

Begin by graphing the function and noticing that nohorizontal line intersects the graph more than once. Thistells you that the inverse of ƒ is itself a function. To findan equation for ƒº1, complete the following steps.

ƒ(x) = �12

�x3 º 2 Write original function.

y = �12

�x3 º 2 Replace ƒ(x) with y.

x = �12

�y3 º 2 Switch x and y.

x + 2 = �12

�y3 Add 2 to each side.

2x + 4 = y3 Multiply each side by 2.

�3 2�x�+� 4� = y Take cube root of each side.

� The inverse function is ƒº1(x) = �3 2�x�+� 4�.

Writing an Inverse Model

ASTRONOMY Near the end of a star’s life the star will eject gas, forming aplanetary nebula. The Ring Nebula is an example of a planetary nebula. The volumeV (in cubic kilometers) of this nebula can be modeled by V = (9.01 ª 1026)t3 wheret is the age (in years) of the nebula. Write the inverse model that gives the age of thenebula as a function of its volume. Then determine the approximate age of the RingNebula given that its volume is about 1.5 ª 1038 cubic kilometers.

SOLUTION

V = (9.01 ª 1026)t3 Write original model.

= t3 Isolate power.

�3 �= t Take cube root of each side.

(1.04 ª 10º9)�3 V� = t Simplify.

To find the age of the nebula, substitute 1.5 ª 1038 for V.

t = (1.04 ª 10º9)�3 V� Write inverse model.

= (1.04 ª 10º9)�3 1�.5� ª� 1�0�38� Substitute for V.

≈ 5500 Use a calculator.

� The Ring Nebula is about 5500 years old.

V��9.01 ª 1026

V��9.01 ª 1026

E X A M P L E 6

E X A M P L E 5

x

1

y

1

ASTRONOMY The Ring Nebula is

part of the constellation Lyra.The radius of the nebula isexpanding at an average rateof about 5.99 �108 kilometersper year.


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1. Explain how to use the horizontal line test to determine if an inverse relation isan inverse function.

2. Describe how the graph of a relation and the graph of its inverse are related.

3. Explain the steps in finding an equation for an inverse function.

Find the inverse relation.

4. 5.

Find an equation for the inverse relation.

6. y = 5x 7. y = 2x º 1 8. y = º�23

�x + 6

Verify that ƒ and g are inverse functions.

9. ƒ(x) = 8x3, g(x) = �x1

2

/3� 10. ƒ(x) = 6x + 3, g(x) = �

16

�x º �12

�

Find the inverse function.

11. ƒ(x) = 3x4, x ≥ 0 12. ƒ(x) = 2x3 + 1

13. The graph of ƒ(x) = º|x| + 1 is shown. Is the inverse ofƒ a function? Explain.

INVERSE RELATIONS Find the inverse relation.

14. 15.

FINDING INVERSES Find an equation for the inverse relation.

16. y = º2x + 5 17. y = 3x º 3 18. y = �12

�x + 6

19. y = º�45

�x + 11 20. y = 11x º 5 21. y = º12x + 7

22. y = 3x º �14

� 23. y = 8x º 13 24. y = º�37

�x + �57

�

VERIFYING INVERSES Verify that ƒ and g are inverse functions.

25. ƒ(x) = x + 7, g(x) = x º 7 26. ƒ(x) = 3x º 1, g(x) = �13�x + �3

1�

27. ƒ(x) = �12�x + 1, g(x) = 2x º 2 28. ƒ(x) = º2x + 4, g(x) = º�

12�x + 2

29. ƒ(x) = 3x3 + 1, g(x) = ��x �

31

��1/330. ƒ(x) = �

13�x2, x ≥ 0; g(x) = (3x)1/2

31. ƒ(x) = , g(x) = �5 7�x�º� 2� 32. ƒ(x) = 256x4, x ≥ 0; g(x) = �4 x��4

x5 + 2�7


GUIDED PRACTICE



Concept Check ✓

Skill Check ✓

STUDENT HELP

Extra Practiceto help you master skills is on p. 949.

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 14–24Example 2: Exs. 25–32Example 3: Exs. 57–59Example 4: Exs. 33–41Example 5: Exs. 42–56Example 6: Exs. 60–62

x 1 2 3 4 5

y º1 º2 º3 º4 º5

x 1 4 1 0 1

y 3 º1 6 º3 9

x 1 º2 4 2 º2

y 0 3 º2 2 º1

x º4 º2 0 2 4

y 2 1 0 1 2

y

x�1

3

Ex. 13


VISUAL THINKING Match the graph with the graph of its inverse.

33. 34. 35.

A. B. C.

INVERSES OF POWER FUNCTIONS Find the inverse power function.

36. ƒ(x) = x7 37. ƒ(x) = ºx6, x ≥ 0 38. ƒ(x) = 3x4, x ≤ 0

39. ƒ(x) = �312�x5 40. ƒ(x) = 10x3 41. ƒ(x) = º�

94�x2, x ≤ 0

INVERSES OF NONLINEAR FUNCTIONS Find the inverse function.

42. ƒ(x) = x3 + 2 43. ƒ(x) = º2x5 + �13� 44. ƒ(x) = 2 º 2x2, x ≤ 0

45. ƒ(x) = �35�x3 º 9 46. ƒ(x) = x4 º �

12�, x ≥ 0 47. ƒ(x) = �

16�x5 + �3

2�

HORIZONTAL LINE TEST Graph the function ƒ. Then use the graph todetermine whether the inverse of ƒ is a function.

48. ƒ(x) = º2x + 3 49. ƒ(x) = x + 3 50. ƒ(x) = x2 + 1

51. ƒ(x) = º3x2 52. ƒ(x) = x3 + 3 53. ƒ(x) = 2x3

54. ƒ(x) = |x| + 2 55. ƒ(x) = (x + 1)(x º 3) 56. ƒ(x) = 6x4 º 9x + 1

57. EXCHANGE RATE The Federal Reserve Bank of New York reportsinternational exchange rates at 12:00 noon each day. On January 20, 1999, the exchange rate for Canada was 1.5226. Therefore, the formula that givesCanadian dollars in terms of United States dollars on that day is

DC = 1.5226DUS

where DC represents Canadian dollars and DUS represents United States dollars. Find the inverse of the function to determine the value of a United States dollar in terms of Canadian dollars on January 20, 1999.

DATA UPDATE of Federal Reserve Bank of New York data at www.mcdougallittell.com

58. TEMPERATURE CONVERSION The formula to convert temperatures fromdegrees Fahrenheit to degrees Celsius is:

C = �59�(F º 32)

Write the inverse of the function, which converts temperatures from degreesCelsius to degrees Fahrenheit. Then find the Fahrenheit temperatures that areequal to 29°C, 10°C, and 0°C.

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1 x

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1

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x1

1

y

x


INVESTMENTBANKER

Investment bankers have a wide variety of jobdescriptions. Some buy andsell international currenciesat reported exchange rates,discussed in Ex. 57.


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59. BOWLING In bowling a handicap is a change in score to adjust fordifferences in players’ abilities. You belong to a bowling league in which eachbowler’s handicap h is determined by his or her average a using this formula:

h = 0.9(200 º a)

(If the bowler’s average is over 200, the handicap is 0.) Find the inverse of thefunction. Then find your average if your handicap is 27.

60. GAMES You and a friend are playing a number-guessing game. You askyour friend to think of a positive number, square the number, multiply the resultby 2, and then add 3. If your friend’s final answer is 53, what was the originalnumber chosen? Use an inverse function in your solution.

61. FISH The weight w (in kilograms) of a hake, atype of fish, is related to its length l (in centimeters)by this function:

w = (9.37 ª 10º6)l3

Find the inverse of the function. Then determinethe approximate length of a hake that weighs 0.679 kilogram. � Source: Fishbyte

62. SHELVES The weight w (in pounds) that can be supported by a shelf madefrom half-inch Douglas fir plywood can be modeled by

w = ��82

d.9��3

where d is the distance (in inches) between the supports for the shelf. Find theinverse of the function. Then find the distance between the supports of a shelfthat can hold a set of encyclopedias weighing 66 pounds.

QUANTITATIVE COMPARISON In Exercises 63 and 64, choose the statementthat is true about the given quantities.





63.

64.

INVERSE FUNCTIONS Complete Exercises 65–68 to explore functions that aretheir own inverses.

65. VISUAL THINKING The functions ƒ(x) = x and g(x) = ºx are their owninverses. Graph each function and explain why this is true.

66. Graph other linear functions that are their own inverses.

67. Write equations of the lines you graphed in Exercise 66.

68. Use your equations from Exercise 67 to find a general formula for a family oflinear equations that are their own inverses.


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EXTRA CHALLENGE


Hake

Column A Column B

ƒº1(3) where ƒ(x) = 6x + 1 ƒº1(º4) where ƒ(x) = º2x + 9

ƒº1(2) where ƒ(x) = º5x3 ƒº1(0) where ƒ(x) = x3 + 14



ABSOLUTE VALUE FUNCTIONS Graph the absolute value function. (Review 2.8 for 7.5)

69. ƒ(x) = |x| º 1 70. ƒ(x) = 2|x| + 7

71. ƒ(x) = |x º 4| + 5 72. ƒ(x) = º3|x + 2| º 7

QUADRATIC FUNCTIONS Graph the quadratic function. (Review 5.1 for 7.5)

73. ƒ(x) = x2 + 2 74. ƒ(x) = (x + 3)2 º 7

75. ƒ(x) = 2(x + 2)2 º 5 76. ƒ(x) = º3(x º 4)2 + 1

SIMPLIFYING EXPRESSIONS Simplify the expression. Assume all variablesare positive. (Review 7.2)

77. �4 2�0� • �4 �45�� 78. ��

19

��1/6 ��19

��1/379.

80. �6 2�x6� 81. 3�7 5� + 2�7 5� 82. �3 2�7�0� + 2�3 1�0�

83. SNACK FOODS Delia, Ruth, and Amy go to the store to buy snacks. Deliabuys 3 bagels and 3 apples. Ruth buys 1 pretzel, 2 bagels, and 3 apples. Amybuys 2 pretzels and 4 bagels. Delia’s bill comes to $3.72, Ruth’s to $5.06, andAmy’s to $6.58. How much does one bagel cost? (Review 3.6)

Let ƒ(x) = 6x2 º x1/2 and g(x) = 2x1/2. Perform the indicated operation andstate the domain. (Lesson 7.3)

1. ƒ(x) + g(x) 2. ƒ(x) º g(x) 3. ƒ(x) • g(x) 4. �ƒg((xx))

�

Let ƒ(x) = 3xº1 and g(x) = x º 8. Perform the indicated operation and state the domain. (Lesson 7.3)

5. ƒ(g(x)) 6. g(ƒ(x)) 7. ƒ(ƒ(x)) 8. g(g(x))

Verify that ƒ and g are inverse functions. (Lesson 7.4)

9. ƒ(x) = 2x º 3, g(x) = �12�x + �

32� 10. ƒ(x) = (x + 1)1/3, g(x) = x3 º 1

Find the inverse function. (Lesson 7.4)

11. ƒ(x) = x + 8 12. ƒ(x) = 2x4, x ≤ 0 13. ƒ(x) = ºx5 + 6

Graph the function ƒ. Then use the graph to determine whether the inverse ofƒ is a function. (Lesson 7.4)

14. ƒ(x) = 3x6 + 2 15. ƒ(x) = º2x5 + 3x º 1 16. ƒ(x) = 6�3 x�+� 4�

17. RIPPLES IN A POND You drop a pebble into a calm pond causing ripples ofconcentric circles. The radius r (in feet) of the outer ripple is given by r(t) = 0.6twhere t is the time (in seconds) after the pebble hits the water. The area A (in square feet) of the outer ripple is given by A(r) = πr2. Use composition of functionsto find the relationship between area and time. Then find the area of the outer ripple after 2 seconds. (Lesson 7.3)

QUIZ 2 Self-Test for Lessons 7.3 and 7.4

(5y)1/5�(5y)6/5

MIXED REVIEW



Using Technology

Graphing Calculator Activity for use with Lesson 7.4ACTIVITY 7.4

Graphing Inverse FunctionsYou can use a graphing calculator to graph inverse functions.

� EXAMPLEUse a graphing calculator to graph the inverse of y = 2x º 5.

� SOLUTIONGraph the original function. Use a viewing window such as º15 ≤ x ≤ 15 and º10 ≤ y ≤ 10.

Use the Draw Inversefeature to graph the inverse.

Display the graphs of the original function and its inverse.

The graph of the function y = 2x º 5 passes the horizontal line test, so you knowthat the inverse of y = 2x º 5 is also a function. You can further verify this fact byobserving that the inverse passes the vertical line test you learned in Lesson 2.1.

� EXERCISES

Graph the function on a graphing calculator. Then use the Draw Inverse featureto graph the function’s inverse in the same viewing window. Is the inverse afunction? Explain.

1. y = 6x + 4 2. y = 0.6x º 2 3. y = 0.4x + 5

4. y = 0.2x2 + 1 5. y = x2 º 4x + 3 6. y = x2 º 3x

7. y = x3 º 4 8. y = x3 + x 9. y = 2.1x3 º 0.4x2 + 1

10. y = |x + 4| 11. y = º|x| + 5.7 12. y = 2|x + 1| º 8

3

2

1

KEYSTROKEHELP

See keystrokes for several models of calculators atwww.mcdougallittell.com

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DRAW POINTS STO2 Line(3:Horizontal4:Vertical5:Tangent(6:DrawF7:Shade(8 DrawInv

DrawInv Y1


Graphing Square Root and Cube Root Functions

GRAPHING RADICAL FUNCTIONS

In Lesson 7.4 you saw the graphs of y = �x� and y = �3x�. These are examples of

Domain: x ≥ 0, Range: y ≥ 0 Domain and range: all real numbers

In this lesson you will learn to graph functions of the form y = a�x�º� h� + k and

y = a�3 x�º� h� + k.

In the activity you may have discovered that the graph of y = a�x� starts at the origin

and passes through the point (1, a). Similarly, the graph of y = a�3x� passes through

the origin and the points (º1, ºa) and (1, a). The following describes how to graphmore general radical functions.

3

y

x

1

(0, 0)(1, 1)

y � 3�x

(�1, �1)

y

x1

1

(0, 0)(1, 1)

y � �x

radical functions.

GOAL 1

7.5 Graphing Square Root and Cube Root Functions 431

Graph square rootand cube root functions.

Use square rootand cube root functions tofind real-life quantities, suchas the power of a race car inEx. 48.

� To solve real-lifeproblems, such as finding theage of an African elephant inExample 6.


GOAL 2

GOAL 1


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To graph y = a�x� º� h� + k or y = a�3 x� º� h� + k, follow these steps.

STEP Sketch the graph of y = a�x� or y = a�3 x�.

STEP Shift the graph h units horizontally and k units vertically.2

1

GRAPHS OF RADICAL FUNCTIONS

Investigating Graphs of Radical FunctionsGraph y = a�x� for a = 2, �

12�, º3, and º1. Use the graph of y = �x�

shown above and the labeled points on the graph as a reference. Describehow a affects the graph.

Graph y = a�3 x� for a = 2, �12�, º3, and º1. Use the graph of y = �3 x�

shown above and the labeled points on the graph as a reference. Describehow a affects the graph.

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DevelopingConcepts

ACTIVITY



Comparing Two Graphs

Describe how to obtain the graph of y = �x�+� 1� º 3 from the graph of y = �x�.

SOLUTION

Note that y = �x�+� 1� º 3 = �x�º� (�º�1�)� + (º3), so h = º1 and k = º3. To obtain

the graph of y = �x�+� 1� º 3, shift the graph of y = �x� left 1 unit and down 3 units.

Graphing a Square Root Function

Graph y = º3�x�º� 2� + 1.

SOLUTION

Sketch the graph of y = º3�x� (shown dashed).Notice that it begins at the origin and passesthrough the point (1, º3).

Note that for y = º3�x�º� 2� + 1, h = 2 and k = 1. So, shift the graph right 2 units and up1 unit. The result is a graph that starts at (2, 1)and passes through the point (3, º2).

Graphing a Cube Root Function

Graph y = 3�3 x�+� 2� º 1.

SOLUTION

Sketch the graph of y = 3�3 x� (shown dashed).Notice that it passes through the origin and thepoints (º1, º3) and (1, 3).

Note that for y = 3�3 x�+� 2� º 1, h = º2 and k = º1. So, shift the graph left 2 units and down 1 unit. The result is a graph that passes throughthe points (º3, º4), (º2, º1), and (º1, 2).

Finding Domain and Range

State the domain and range of the function in (a) Example 2 and (b) Example 3.

SOLUTION

a. From the graph of y = º3�x�º� 2� + 1 in Example 2, you can see that thedomain of the function is x ≥ 2 and the range of the function is y ≤ 1.

b. From the graph of y = 3�3x�+� 2� º 1 in Example 3, you can see that the domain

and range of the function are both all real numbers.

E X A M P L E 4

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y

x1

1 (0, 0)(2, 1)

(1, �3)

y � �3�x

y � �3�x � 2 � 1

(3, �2)

y � 3 3�x

(0, 0)

y

x1

1

(1, 3)

(�1, �3)

(�1, 2)

(�3, �4)

y � 3 3�x � 2 � 1

(�2, �1)

Skills ReviewFor help withtransformations, see p. 921.

STUDENT HELP



USING RADICAL FUNCTIONS IN REAL LIFE

When you use radical functions in real life, the domain is understood to be restrictedto the values that make sense in the real-life situation.

Modeling with a Square Root Function

AMUSEMENT PARKS At an amusement parka ride called the rotor is a cylindrical room that

spins around. The riders stand against the circularwall. When the rotor reaches the necessary speed,the floor drops out and the centrifugal force keepsthe riders pinned to the wall.

The model that gives the speed s (in meters persecond) necessary to keep a person pinned to thewall is

s = 4.95�r�

where r is the radius (in meters) of the rotor. Use a graphing calculator to graph themodel. Then use the graph to estimate the radius of a rotor that spins at a speed of8 meters per second.

SOLUTION

Graph y = 4.95�x� and y = 8. Choose a viewingwindow that shows the point where the graphsintersect. Then use the Intersect feature to find the x-coordinate of that point. You get x ≈ 2.61.

� The radius is about 2.61 meters.

Modeling with a Cube Root Function

Biologists have discovered that the shoulder height h (in centimeters) of a maleAfrican elephant can be modeled by

h = 62.5�3 t� + 75.8

where t is the age (in years) of the elephant. Use a graphing calculator to graph themodel. Then use the graph to estimate the age of an elephant whose shoulder heightis 200 centimeters. � Source: Elephants

SOLUTION

Graph y = 62.5�3 x� + 75.8 and y = 200. Choose aviewing window that shows the point where thegraphs intersect. Then use the Intersect feature to findthe x-coordinate of that point. You get x ≈ 7.85.

� The elephant is about 8 years old.

E X A M P L E 6

E X A M P L E 5

GOAL 2

IntersectionX=2.6119784 Y=8


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Biology

AMUSEMENTRIDE DESIGNER

An amusement ride designeruses math and science toensure the safety of therides. Most amusement ridedesigners have a degree inmechanical engineering.


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1. Complete this statement: Square root functions and cube root functions areexamples of ��? functions.

2. ERROR ANALYSIS Explain why thegraph shown at the near right is not the

graph of y = �x�º� 1� + 2.

3. ERROR ANALYSIS Explain why thegraph shown at the far right is not the

graph of y = �3 x�+� 2� º 3.

Describe how to obtain the graph of g from the graph of ƒ.

4. g(x) = �x�+� 5�, ƒ(x) = �x� 5. g(x) = �3 x� º 10, ƒ(x) = �3 x�

Graph the function. Then state the domain and range.

6. y = º�x� 7. y = �x�+� 1� 8. y = �x�º� 2� 9. y = 2�x�+� 3� º 1

10. y = �23��3 x� 11. y = �3 x� º 6 12. y = �3 x�+� 5� 13. y = º3�3 x�º� 7� º 4

14. ELEPHANTS Look back at Example 6. Use a graphing calculator to graphthe model. Then use the graph to estimate the age of an elephant whose

shoulder height is 250 centimeters.

COMPARING GRAPHS Describe how to obtain the graph of g from the graphof ƒ.

15. g(x) = �x�+� 1�4�, ƒ(x) = �x� 16. g(x) = 5�x�º� 1�0� º 3, ƒ(x) = 5�x�

17. g(x) = º�3 x� º 10, ƒ(x) = º�3 x� 18. g(x) = �3 x�+� 6� º 5, ƒ(x) = �3 x�

MATCHING GRAPHS Match the function with its graph.

19. y = �x� º 1 20. y = �x�+� 1� 21. y = �x�+� 1�º1

A. B. C.

SQUARE ROOT FUNCTIONS Graph the function. Then state the domain and range.

22. y = º5�x� 23. y = �13��x� 24. y = x1/2 + �4

1�

25. y = x1/2 º 2 26. y = �x�+� 6� 27. y = (x º 7)1/2

28. y = (x º 1)1/2 + 7 29. y = 2�x�+� 5� º 1 30. y = º�25��x�º� 3� º 2

(0, 0)

(�1, �1)

2

y

x1

(1, 0)

(0, �1)

y

x2

1

(�1, 0)

y

x1

2

(0, 1)


GUIDED PRACTICE



Concept Check ✓

Skill Check ✓

y

-1x1

(2, -3)

(3, -2)

(1, -4)

y

x1

1(2, 1)(3, 2)


STUDENT HELP

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 15–18Example 2: Exs. 19–30Example 3: Exs. 31–39Example 4: Exs. 22–45Example 5: Exs. 46, 47Example 6: Exs. 48, 49

Ex. 2 Ex. 3



CUBE ROOT FUNCTIONS Graph the function. Then state the domain and range.

31. y = �12��3 x� 32. y = º2x1/3 33. y = �3 x� º 7

34. y = �3 x� + �34� 35. y = �3 x�º� 5� 36. y = �x + �

23��1/3

37. y = �15�x1/3 º 2 38. y = º3�3 x�+� 4� 39. y = 2�3 x�º� 4� + 3

CRITICAL THINKING Find the domain and range of the function withoutgraphing. Explain how you found your solution.

40. y = �x�º� 1�3� 41. y = 2�x� º 2 42. y = º�x�º� 3� º 7

43. y = �3 x�+� 8� 44. y = º�23� �3 x� º 5 45. y = 4�3 x�+� 4� + 7

GRAPHING MODELS In Exercises 46–49, use a graphing calculator tograph the models. Then use the Intersect feature to solve the problems.

46. OCEAN DISTANCES When you look at the ocean, the distance d (in miles)

you can see to the horizon can be modeled by d = 1.22�a� where a is youraltitude (in feet above sea level). Graph the model. Then determine at whataltitude you can see 10 miles. � Source: Mathematics in Everyday Things

47. In a right circular cone with a slant height of 1 unit, the radius r of the cone is given

by r = �S� +� �π4�� º �

12� where S is the surface area of

the cone. Graph the model. Then find the surface area of a right circular cone with a slant height of 1 unit and

a radius of �12� unit.

48. RACING Drag racing is an acceleration contest over a distance of a quartermile. For a given total weight, the speed of a car at the end of the race is afunction of the car’s power. For a total weight of 3500 pounds, the speed s

(in miles per hour) can be modeled by s = 14.8�3 p� where p is the power (in horsepower). Graph the model. Then determine the power of a car thatreaches a speed of 100 miles per hour. � Source: The Physics of Sports

49. STORMS AT SEA The fetch ƒ (in nautical miles) of the wind at sea is thedistance over which the wind is blowing. The minimum fetch required to create afully developed storm can be modeled by s = 3.1�3 f�+� 1�0� + 11.1 where s is thespeed (in knots) of the wind. Graph the model. Then determine the minimum fetchrequired to create a fully developed storm if the wind speed is 25 knots. � Source: Oceanography

1��π�

GEOMETRY CONNECTION

1

12

wind

fetch

limitsof storm

wave direction

COAST GUARDMembers of the

Coast Guard have a varietyof responsibilities. Someparticipate in search andrescue missions that involverescuing people caught instorms at sea, discussed in Ex. 49.


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www.mcdougallittell.comfor help with problemsolving in Exs. 40–45.

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50. MULTI-STEP PROBLEM Follow the steps below to graph radical functions ofthe form y = ƒ(ºx).

a. Graph ƒ1(x) = �x� and ƒ2(x) = �º�x�. How are the graphs related?

b. Graph g1(x) = �3 x� and g2(x) = �3 º�x�. How are the graphs related?

c. Graph ƒ3(x) = �º�(x� º� 2�)� º 4 and g3(x) = 2�3 º�(x� +� 1�)� + 5 using what youlearned from parts (a) and (b) and what you know about the effects of a, h,

and k on the graphs of y = a�x�º� h� + k and y = a�3 x�º� h� + k.

d. Writing Describe the steps for graphing a function of the form

ƒ(x) = a�º�(x� º� h�)� + k or g(x) = a�3 º�(x� º� h�)� + k.

ANALYZING GRAPHS Write an equation for the function whose graph is shown.

51. 52. 53.

SOLVING EQUATIONS Solve the equation. (Review 5.3 for 7.6)

54. 2x2 = 32 55. (x + 7)2 = 10 56. 9x2 + 3 = 5

57. �12�x2 º 5 = 13 58. �

14�(x + 6)2 = 22 59. 2(x º 0.25)2 = 16.5

SPECIAL PRODUCTS Find the product. (Review 6.3 for 7.6)

60. (x + 4)2 61. (x º 9y)2 62. (2x3 + 7)2

63. (º3x + 4y4)2 64. (6 º 5x)2 65. (º1 º 2x2)2

COMPOSITION OF FUNCTIONS Find ƒ(g(x)) and g(ƒ(x)). (Review 7.3)

66. ƒ(x) = x + 7, g(x) = 2x 67. ƒ(x) = 2x + 1, g(x) = x º 3

68. ƒ(x) = x2 º 1, g(x) = x + 2 69. ƒ(x) = x2 + 7, g(x) = 3x º 3

70. MANDELBROT SET To determine whether a complex number c belongs tothe Mandelbrot set, consider the function ƒ(z) = z2 + c and the infinite list ofcomplex numbers z0 = 0, z1 = ƒ(z0), z2 = ƒ(z1), z3 = ƒ(z2), . . . .

• If the absolute values |z0|, |z1|, |z2|, |z3|, . . . are all less than some fixednumber N, then c belongs to the Mandelbrot set.

• If the absolute values |z0|, |z1|, |z2|, |z3|, . . . become infinitely large,then c does not belong to the Mandelbrot set.

Tell whether the complex number c belongs to the Mandelbrot set. (Review 5.4)

a. c = 3i b. c = 2 + 2i c. c = 6

MIXED REVIEW

1

1

y

x

y

x1

1

y

x1

1


TestPreparation

★★ Challenge

EXTRA CHALLENGE



Solving Radical EquationsSOLVING A RADICAL EQUATION

To solve a radical equation—an equation that contains radicals or rationalexponents—you need to eliminate the radicals or rational exponents and obtain apolynomial equation. The key step is to raise each side of the equation to thesame power.

If a = b, then an = bn. Powers property of equality

Then solve the new equation using standard procedures. Before raising each side ofan equation to the same power, you should isolate the radical expression on one sideof the equation.

Solving a Simple Radical Equation

Solve �3 x� º 4 = 0.

SOLUTION

�3 x� º 4 = 0 Write original equation.

�3 x� = 4 Isolate radical.

(�3 x�)3 = 43 Cube each side.

x = 64 Simplify.

� The solution is 64. Check this in the original equation.

Solving an Equation with Rational Exponents

Solve 2x3/2 = 250.

SOLUTION

Because x is raised to the �32� power, you should isolate the power and then raise each

side of the equation to the �23� power ��

23� is the reciprocal of �

32��.

2x3/2 = 250 Write original equation.

x3/2 = 125 Isolate power.

(x3/2)2/3 = 1252/3 Raise each side to }23} power.

x = (1251/3)2 Apply properties of roots.

x = 52 = 25 Simplify.

� The solution is 25. Check this in the original equation.

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E X A M P L E 1

GOAL 1

7.6 Solving Radical Equations 437

Solve equationsthat contain radicals orrational exponents.

Use radicalequations to solve real-lifeproblems, such as determin-ing wind speeds that corre-spond to the Beaufort windscale in Example 6.

� To solve real-lifeproblems, such as determin-ing which boats satisfy therule for competing in theAmerica’s Cup sailboat racein Ex. 68.


GOAL 2

GOAL 1


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STUDENT HELP

Study TipTo solve an equation ofthe form xm/n = k wherek is a constant, raise bothsides of the equation tothe �m

n� power, because

(xm/n)n/m = x1 = x.



Solving an Equation with One Radical

Solve �4�x�º� 7� + 2 = 5.

SOLUTION

�4�x�º� 7� + 2 = 5 Write original equation.

�4�x�º� 7� = 3 Isolate radical.

(�4�x�º� 7�)2 = 32 Square each side.

4x º 7 = 9 Simplify.

4x = 16 Add 7 to each side.

x = 4 Divide each side by 4.

✓ CHECK Check x = 4 in the original equation.

�4�x� º� 7� + 2 = 5 Write original equation.

�4�(4�)�º� 7� · 3 Substitute 4 for x.

�9� · 3 Simplify.

3 = 3 ✓ Solution checks.

� The solution is 4.

. . . . . . . . . .

Some equations have two radical expressions. Before raising both sides to the samepower, you should rewrite the equation so that each side of the equation has only oneradical expression.

Solving an Equation with Two Radicals

Solve �3�x�+� 2� º 2�x� = 0.

SOLUTION

�3�x�+� 2� º 2�x� = 0 Write original equation.

�3�x�+� 2� = 2�x� Add 2�x� to each side.

(�3�x�+� 2� )2 = (2�x�)2 Square each side.

3x + 2 = 4x Simplify.

2 = x Solve for x.


�3�x� +� 2� º 2�x� = 0 Write original equation.

�3�(2�)�+� 2� º 2�2� · 0 Substitute 2 for x.

2�2� º 2�2� · 0 Simplify.


� The solution is 2.

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If you try to solve �x� = º1 by squaring both sides, you get x = 1. But x = 1 is not a valid solution of the original equation. This is an example of an (or false) Raising both sides of an equation to the same power mayintroduce extraneous solutions. So, when you use this procedure it is critical that you check each solution in the original equation.

An Equation with an Extraneous Solution

Solve x º 4 = �2�x�.

SOLUTION

x º 4 = �2�x� Write original equation.

(x º 4)2 = (�2�x�)2 Square each side.

x2 º 8x + 16 = 2x Expand left side; simplify right side.

x2 º 10x + 16 = 0 Write in standard form.

(x º 2)(x º 8) = 0 Factor.

x º 2 = 0 or x º 8 = 0 Zero product property

x = 2 or x = 8 Simplify.



2 º 4 · �2�(2�)� Substitute 2 for x.

º2 · �4� Simplify.

º2 ≠ 2 Solution does not check.



8 º 4 · �2�(8�)� Substitute 8 for x.

4 · �1�6� Simplify.


� The only solution is 8.

. . . . . . . . . .

If you graph each side of the equation in Example 5,as shown, you can see that the graphs of y = x º 4and y = �2�x� intersect only at x = 8. This confirmsthat x = 8 is a solution of the equation, but that x = 2 is not.

In general, all, some, or none of the apparentsolutions of a radical equation can be extraneous.When all of the apparent solutions of a radical equation are extraneous, the equation has no solution.

E X A M P L E 5

solution.extraneous

IntersectionX=8 Y=4

Look Back For help with factoring,see p. 256.

STUDENT HELP



SOLVING RADICAL EQUATIONS IN REAL LIFE

Using a Radical Model

BEAUFORT WIND SCALE The Beaufort wind scale was devised to measure windspeed. The Beaufort numbers B, which range from 0 to 12, can be modeled by B = 1.69�s�+� 4�.4�5� º 3.49 where s is the speed (in miles per hour) of the wind. Find the wind speed that corresponds to the Beaufort number B = 11.

SOLUTION

B = 1.69�s�+� 4�.4�5� º 3.49 Write model.

11 = 1.69�s�+� 4�.4�5� º 3.49 Substitute 11 for B.

14.49 = 1.69�s�+� 4�.4�5� Add 3.49 to each side.

8.57 ≈ �s�+� 4�.4�5� Divide each side by 1.69.

73.4 ≈ s + 4.45 Square each side.

69.0 ≈ s Subtract 4.45 from each side.

� The wind speed is about 69 miles per hour.

✓ ALGEBRAIC CHECK Substitute 69 for s into themodel and evaluate.

1.69�6�9� +� 4�.4�5� º 3.49 ≈ 1.69(8.57) º 3.49≈ 11 ✓

✓ GRAPHIC CHECK You can use a graphing calculatorto graph the model, and then use the Intersect featureto check that x ≈ 69 when y = 11.

E X A M P L E 6

GOAL 2


Beaufort Wind Scale

Beaufortnumber Force of wind Effects of wind

0 Calm Smoke rises vertically.

1 Light air Direction shown by smoke.

2 Light breeze Leaves rustle; wind felt on face.

3 Gentle breeze Leaves move; flags extend.

4 Moderate breeze Small branches sway; paper blown about.

5 Fresh breeze Small trees sway.

6 Strong breeze Large branches sway; umbrellas difficult to use.

7 Moderate gale Large trees sway; walking difficult.

8 Fresh gale Twigs break; walking hindered.

9 Strong gale Branches scattered about; slight damage to buildings.

10 Whole gale Trees uprooted; severe damage to buildings.

11 Storm Widespread damage.

12 Hurricane Devastation.

BEAUFORT WIND SCALE

The Beaufort wind scalewas developed by Rear-Admiral Sir Francis Beaufortin 1805 so that sailors coulddetect approaching storms.Today the scale is usedmainly by meteorologists.


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1. What is an extraneous solution?

2. Marcy began solving x2/3 = 5 by cubing each side. What will she have to donext? What could she have done to solve the equation in just one step?

3. Zach was asked to solve �5�x�º� 2� º �7�x�º� 4� = 0. His first step was to squareeach side. While trying to isolate x, he gave up in frustration. What could Zachhave done to avoid this situation?

Solve the rational exponent equation. Check for extraneous solutions.

4. 3x1/4 = 4 5. (2x + 7)3/2 = 27 6. x4/3 + 9 = 25

7. 4x2/3 º 6 = 10 8. 5(x º 8)3/4 = 40 9. (x + 9)5/2 º 1 = 31

Solve the radical equation. Check for extraneous solutions.

10. �4 x� = 3 11. �3 3�x� + 6 = 10 12. �5 2�x�+� 1� + 5 = 9

13. �x�º� 2� = x º 2 14. �3 x�+� 4� = �3 2�x�º� 5� 15. 6�x� º �x�º� 1� = 0

16. BEAUFORT WIND SCALE Use the information in Example 6 to determinethe wind speed that corresponds to the Beaufort number B = 2.

CHECKING SOLUTIONS Check whether the given x-value is a solution of the equation.

17. �x� º 3 = 6; x = 81 18. 4(x º 5)1/2 = 28; x = 12

19. (x + 7)3/2 º 20 = 7; x = 2 20. �3 4�x� + 11 = 5; x = º54

21. 2�5�x�+� 4� + 10 = 10; x = 0 22. �4�x�º� 3� º �3�x� = 0; x = 3

SOLVING RATIONAL EXPONENT EQUATIONS Solve the equation. Check for extraneous solutions.

23. x5/2 = 32 24. x1/3 º �25� = 0 25. x2/3 + 15 = 24

26. º�12�x1/5 = 10 27. 4x3/4 = 108 28. (x º 4)3/2 = º6

29. (2x + 5)1/2 = 4 30. 3(x + 1)4/3 = 48 31. º(x º 5)1/4 + �73� = 2

SOLVING RADICAL EQUATIONS Solve the equation. Check for extraneoussolutions.

32. �x� = �19� 33. �3

x� + 10 = 16 34. �4 2�x� º 13 = º9

35. �x�+� 5�6� = 16 36. �3 x�+� 4�0� = º5 37. �6�x�º� 5� + 10 = 3

38. �25��1�0�x�+� 6� = 12 39. 2�7�x�+� 4� º 1 = 7 40. º2�5 2�x�º� 1� + 4 = 0

41. x º 12 = �1�6�x� 42. �4 x4� +� 1� = 3x 43. �x2� +� 5� = x + 3

44. �3 x� = x º 6 45. �8�x�+� 1� = x + 2 46. �2�x�+�� = x + 5�61�6


GUIDED PRACTICE


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STUDENT HELP


32–46Example 2: Exs. 17–22,

23–31Example 3: Exs. 17–22,

32–46Example 4: Exs. 17–22,

47–54Example 5: Exs. 23–54Example 6: Exs. 63–69



SOLVING EQUATIONS WITH TWO RADICALS Solve the equation. Check forextraneous solutions.

47. �2�x�º� 1� = �x�+� 4� 48. �4 6�x�º� 5� = �4 x�+� 1�0�

49. º�8�x�+� �43�� = �2�x�+� �

13�� 50. 2�3 1�0� º� 3�x� = �3 2� º� x�

51. �4 2�x� + �4 x�+� 3� = 0 52. �x�º� 6� º ��13��x� = 0

53. �2�x�+� 1�0� º 2�x� = 0 54. �3 2�x�+� 1�5� º �32��3 x� = 0

SOLVING EQUATIONS Use the Intersect feature on a graphing calculatorto solve the equation.

55. �34�x1/3 = º2 56. 2(x + 19)2/5 º 1 = 17

57. (3.5x + 1)2/7 = (6.4x + 0.7)2/7 58. ��15�x�3/4

= x º �83

�

59. �6�.7�x�+� 1�4� = 9.4 60. �37�0� º� 2�x� º 10 = º6

61. �4 x�º� �16�� = 2�4 3�x� 62. �1�.1�x�+� 2�.4� = 19x º 4.2

63. NAILS The length l (in inches) of a standard nail can be modeled by

l = 54d3/2

where d is the diameter (in inches) of the nail. What is the diameter of a standardnail that is 3 inches long?

64. Scientists have found that the body mass m (inkilograms) of a dinosaur that walked on two feet can be modeled by

m = (1.6 ª 10º4)C273/100

where C is the circumference (in millimeters) of the dinosaur’s femur. Scientistshave estimated that the mass of a Tyrannosaurus rex might have been4500 kilograms. What size femur would have led them to this conclusion? � Source: The Zoological Society of London

65. WOMEN IN MEDICINE For 1970 through 1995, the percent p of Doctor ofMedicine (MD) degrees earned each year by women can be modeled by

p = (0.867t2 + 39.2t + 57.1)1/2

where t is the number of years since 1970. In what year were about 36% of thedegrees earned by women? � Source: Statistical Abstract of the United States

66. PLUMB BOBS You work for a company thatmanufactures plumb bobs. The same mold is usedto cast plumb bobs of different sizes. The equation

h = 1.5�3t�, 0 ≤ h ≤ 3

models the relationship between the height h(in inches) of the plumb bob and the time t (inseconds) that metal alloy is poured into the mold.How long should you pour the alloy into the moldto cast a plumb bob with a height of 2 inches?

SCIENCE CONNECTION

DR. ALEXACANADY was the

first African-Americanwoman to become aneurosurgeon in the UnitedStates. She received her MDdegree, discussed in Ex. 65,in 1975.

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67. BEAUFORT WIND SCALE Recall from Example 6 that the Beaufortnumber B from the Beaufort wind scale can be modeled by

B = 1.69�s�+� 4�.4�5� º 3.49

where s is the speed (in miles per hour) of the wind. Find the wind speed thatcorresponds to the Beaufort number B = 7.

68. AMERICA’S CUP In order to compete in the America’s Cup sailboat race, a boat must satisfy the rule

≤ 24

where l is the length (in meters) of the boat, s is the area (in square meters) of thesails, and d is the volume (in cubic meters) of water displaced by the boat. If aboat has a length of 20 meters and a sail area of 300 square meters, what is theminimum allowable value for d? � Source: America’s Cup

69. You are trying todetermine the height of a truncated pyramid thatcannot be measured directly. The height h and slantheight l of a truncated pyramid are related by theformula

l = �h�2�+�� 14��(��b�2��º�� b��1)��2��

where b1 and b2 are the lengths of the upper and lower bases of the pyramid,respectively. If l = 5, b1 = 2, and b2 = 4, what is the height of the pyramid?

70. CRITICAL THINKING Look back at Example 5. Solve x º 4 = º�2�x� instead of x º 4 = �2�x�. How does changing �2�x� to º�2�x� change the solution(s) of the equation?

71. MULTIPLE CHOICE What is the solution of �6�x�º� 4� = 3?

¡A º�16� ¡B �

56� ¡C �

76� ¡D �

53� ¡E �

163�

72. MULTIPLE CHOICE What is (are) the solution(s) of �2�x�º� 3� = �12�x?

¡A 2 ¡B 2, 6 ¡C �178� ¡D �

241� ¡E none

73. MULTIPLE CHOICE What is the solution of �3 x�º� 7� = �3 �34��x�+� 1�?

¡A º6 ¡B º�274� ¡C º4 ¡D 2 ¡E 32

SOLVING EQUATIONS WITH TWO RADICALS Solve the equation. Check forextraneous solutions. (Hint: To solve these equations you will need to squareeach side of the equation two separate times.)

74. �x�+� 5� = 5 º �x� 75. �2�x�+� 3� = 3 º �2�x�

76. �x�+� 3� º �x�º� 1� = 1 77. �2�x�+� 4� + �3�x�º� 5� = 4

78. �3�x�º� 2� = 1 + �2�x�º� 3� 79. �12��2�x�º� 5� º �

12��3�x�+� 4� = 1

GEOMETRY CONNECTION

l + 1.25�s� º 9.8�3d�

��0.679

4

h

TestPreparation

EXTRA CHALLENGE


★★ Challenge


USING ORDER OF OPERATIONS Evaluate the expression. (Review 1.2 for 7.7)

80. 6 + 24 ÷ 3 81. 3 • 5 + 10 ÷ 2 82. 27 º 4 • 16 ÷ 8

83. 2 º (10 • 2)2 ÷ 5 84. 8 + (3 • 10) ÷ 6 º 1 85. 11 º 8 ÷ 2 + 48 ÷ 4

USING GRAPHS Graph the polynomial function. Identify the x-intercepts, local maximums, and local minimums. (Review 6.8)

86. ƒ(x) = x3 º 4x2 + 3 87. ƒ(x) = 3x3 º 2.5x2 + 1.25x + 6

88. ƒ(x) = �12�x4 º �

12� 89. ƒ(x) = x5 + x3 º 6x

90. PRINTING RATES The cost C (in dollars) of printing x announcements (inhundreds) is given by the function shown. Graph the function. (Review 2.7)

62 + 22(x º 1), if 1 ≤ x ≤ 5150 + 14(x º 5), if x > 5C =

MIXED REVIEW

Tsunamis

THENTHEN IN AUGUST OF 1883, a volcano erupted on the island of Krakatau,Indonesia. The eruption caused a tsunami (a type of wave) to formand travel into the Indian Ocean and into the Java Sea. The speeds (in kilometers per hour) that a tsunami travels can be modeledby s = 356�d� where d is the depth (in kilometers) of the water.

1. A tsunami from Krakatau hit Jakarta traveling about 60 kilometers per hour. What is the average depth of thewater between Krakatau and Jakarta?

2. After 15 hours and 12 minutes a tsunami from Krakatau hitPort Elizabeth, South Africa, 7546 kilometers away. Find theaverage speed of the tsunami.

3. Based on your answer to Exercise 2, what is the averagedepth of the Indian Ocean between Krakatau and Port Elizabeth?

AFTER A TRAGIC TSUNAMI hit the Aleutian Islands in 1946, scientists began work ona tsunami warning system. Today that system is operated 24 hours a day at theHonolulu Observatory and effectively warns people when a tsunami might arrive.


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ASIA

AFRICA

AUSTRALIA

PacificOcean

IndianOcean

KrakatauJakarta

Port Elizabeth

Famous tsunami artcreated by Hokusai.

Krakatau erupts.

Prototype of tsunamireal-time reportingsystem developed.

Alaskan earthquakecauses Pacific-widetsunami.

1883

1957

1995

c. 1800


7.7 Statistics and Statistical Graphs 445

Statistics and Statistical GraphsMEASURES OF CENTRAL TENDENCY AND DISPERSION

In this lesson you will use the following two data sets. They show the free-throwpercentages for the players in the Women’s National Basketball Association (WNBA)for 1998.

are numerical values used to summarize and compare sets of data. Thefollowing are three commonly used statistics.

1. The or average, of n numbers is the sum of the numbers divided by n.The mean is denoted by x�, which is read as “x-bar.” For the data x1, x2, . . . , xn,

the mean is x� = .

2. The of n numbers is the middle number when the numbers are writtenin order. (If n is even, the median is the mean of the two middle numbers.)

3. The of n numbers is the number or numbers that occur most frequently.There may be one mode, no mode, or more than one mode.

Finding Measures of Central Tendency

Find the mean, median, and mode of the two data sets listed above.

SOLUTION

EASTERN CONFERENCE: Mean: x� = = �395

37

1� ≈ 69.0

Median: 69 Mode: 63

WESTERN CONFERENCE: Mean: x� = = �415

08

6� ≈ 70.8

Median: 71 Modes: 66, 70, 71, 74

All three measures of central tendency for the Western Conference are greater thanthose for the Eastern Conference. So, the Western Conference has better free-throwpercentages overall.

36 + 50 + . . . + 100��58

46 + 47 + . . . + 100��57

E X A M P L E 1

mode

median

x1+ x2 + . . . + xn��n

mean,

measures of central tendencyStatistics

GOAL 1

Use measures ofcentral tendency andmeasures of dispersion todescribe data sets.

Use box-and-whisker plots and histogramsto represent data graphically,as applied in Exs. 40 –42.

� To use statistics andstatistical graphs to analyzereal-life data sets, such asthe free-throw percentagesfor the players in the WNBAin Examples 1–6.


GOAL 2

GOAL 1


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Women’s National Basketball Association Free-Throw Percentages

Eastern Conference Western Conference

46, 47, 48, 50, 50, 50, 52, 53, 55, 57, 36, 50, 50, 56, 56, 57, 57, 58, 61, 61,57, 58, 60, 61, 61, 62, 63, 63, 63, 63, 62, 62, 63, 63, 64, 64, 65, 66, 66, 66,63, 63, 63, 64, 64, 67, 67, 67, 69, 71, 66, 67, 69, 69, 70, 70, 70, 70, 71, 71, 72, 72, 72, 72, 73, 75, 75, 75, 75, 75, 71, 71, 72, 72, 73, 73, 74, 74, 74, 74,76, 77, 78, 79, 79, 80, 81, 81, 82, 82, 75, 76, 76, 76, 77, 77, 78, 80, 81, 83, 83, 83, 85, 89, 91, 92, 100 83, 83, 85, 85, 87, 100, 100, 100

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Measures of central tendency tell you what the center of the data is. Other commonlyused statistics are called They tell you how spread out thedata are. One simple measure of dispersion is the which is the differencebetween the greatest and least data values.

Finding Ranges of Data Sets

The ranges of the free-throw percentages in the two data sets on the previous page are:

EASTERN CONFERENCE: Range = 100 º 46 = 54

WESTERN CONFERENCE: Range = 100 º 36 = 64

Because the Western Conference’s range of free-throw percentages is greater, its free-throw percentages are more spread out.

. . . . . . . . . .

Another measure of dispersion is which describes the typicaldifference (or deviation) between the mean and a data value.

Finding Standard Deviations of Data Sets

The standard deviations of the free-throw percentages in the two data sets on theprevious page are:

EASTERN CONFERENCE: � ≈ ��≈ ��

8�56�760��

≈ �1�5�2�

≈ 12.3

WESTERN CONFERENCE: � ≈ ��≈ ��

7�59�810��

≈ �1�3�6�

≈ 11.7

Because the Eastern Conference’s standard deviation is greater, its free-throwpercentages are more spread out about the mean.

(36 º 70.8)2 + (50 º 70.8)2 + . . . + (100 º 70.8)2��58

(46 º 69.0)2 + (47 º 69.0)2 + . . . + (100 º 69.0)2��57

E X A M P L E 3

standard deviation,

E X A M P L E 2

range,measures of dispersion.

The standard deviation ß (read as “sigma” ) of x1, x2, . . . , xn is:

ß = ��(x1 º xÆ )2 + (x2 º xÆ )2 + . . . + (xn º xÆ )2

��n

STANDARD DEVIATION OF A SET OF DATA

STUDENT HELP



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USING STATISTICAL GRAPHS

Although statistics are useful in describing a data set, sometimes a graph of the datacan be more informative. One type of statistical graph is a The “box” encloses the middle half of the data set and the “whiskers” extend to theminimum and maximum data values.

The median divides the data set into two halves. The is the medianof the lower half, and the is the median of the upper half. You canuse the following steps to draw a box-and-whisker plot.

1. Order the data from least to greatest.

2. Find the minimum and maximum values.

3. Find the median.

4. Find the lower and upper quartiles.

5. Plot these five numbers below a number line.

6. Draw the box, the whiskers, and a line segment through the median.

Drawing Box-and-Whisker Plots

Draw a box-and-whisker plot of each data set on page 445.

SOLUTION

EASTERN CONFERENCE

The minimum is 46 and the maximum is 100. The median is 69. The lower quartile is 61 and the upper quartile is 78.5.

WESTERN CONFERENCE

The minimum is 36 and the maximum is 100. The median is 71. The lower quartile is 64 and the upper quartile is 76.

Like the computations in Examples 2 and 3, the box-and-whisker plots show you that the Western Conference’s free-throw percentages are more spread out overall(comparing the entire plots) and that the Eastern Conference’s free-throw percentagesare more spread out about the mean (comparing the boxes).

E X A M P L E 4

upper quartilelower quartile

box-and-whisker plot.

GOAL 2

0 20 40 10060 80

39 565 72 140

maximumupperquartile

medianlowerquartile

minimum

120 140 160

50 907030

46 61 69 100

110

78.5

50 907030

36 64 71 100

110

76

SANDYBRONDELLO was

ranked first in the WNBA in1998 for free-throw percent-age (among players whoattempted at least 10 freethrows). As a player for theDetroit Shock, she made 96out of 104 free throws for afree-throw percentage of 92.

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Another way to display numerical data is with a special type of bar graph called aIn a histogram data are grouped into intervals of equal width. The

number of data values in each interval is the of the interval. To draw ahistogram, begin by making a which shows the frequencyof each interval.

Making Frequency Distributions

Make a frequency distribution of each data set on page 445. Use seven intervalsbeginning with the interval 31–40.

SOLUTION Begin by writing the seven intervals. Then tally the data values byinterval. Finally, count the tally marks to get the frequencies.

Drawing Histograms

Draw a histogram of each data set on page 445.

SOLUTION Use the frequency distributions in Example 5. Draw a horizontal axis,divide it into seven equal sections, and label the sections with the intervals. Thendraw a vertical axis for measuring the frequencies. Finally, draw bars of appropriateheights to represent the frequencies of the intervals.

E X A M P L E 6

E X A M P L E 5

frequency distribution,frequency

histogram.

Eastern Conference Western Conference

Interval Tally Frequency Interval Tally Frequency

31–40 0 31–40 1

41–50 6 41–50 2

51–60 7 51–60 5

61–70 16 61–70 20

71–80 17 71–80 20

81–90 8 81–90 7

91–100 3 91–100 3

Num

ber o

f pla

yers

41–50

51–60

61–70

71–80

81–90

91–10

0

Eastern Conference

25

20

15

10

0

Free-throw percentages

31–40

5

Num

ber o

f pla

yers

41–50

51–60

61–70

71–80

81–90

91–10

0

Western Conference

25

20

15

10

0

Free-throw percentages

31–40

5

Skills Review For help with statisticalgraphs, see p. 934.

STUDENT HELP



1. Define mean, median, mode, and range of a set of n numbers.

2. Give an example of two sets of four numbers, each with a mean of 5. Choose thenumbers so that one set has a range of 3 and the other set has a range of 7. Whichhas the greater standard deviation?

3. The following box-and-whisker plots represent two sets of data. Which data sethas the greater range? Explain.

A. B.

HISTORY SCORES In Exercises 4–9, use the following data set of scoresreceived by students on a history exam.

68, 72, 76, 81, 84, 86, 86, 86, 89, 91, 95, 99

4. Find the mean, median, and mode of the data.

5. Find the range of the data.

6. Find the standard deviation of the data.

7. Draw a box-and-whisker plot of the data.

8. Make a frequency distribution of the data. Use five intervals beginning with 61–68.

9. Draw a histogram of the data.

MEASURES OF CENTRAL TENDENCY Find the mean, median, and mode ofthe data set.

10. 6, 7, 9, 9, 9, 10 11. 43, 46, 47, 47, 51, 54, 59

12. 88, 83, 91, 82, 78, 81, 91, 91 13. 220, 250, 210, 290, 310, 230, 230

14. 2.9, 2.1, 2.6, 2.9, 3.0, 2.5, 3.4 15. 0, 0, 0.1, 0.2, 0.3, 0.5, 0.5, 0.6, 1.0

MEASURES OF DISPERSION Find the range and standard deviation of thedata set.

16. 10, 20, 30, 40, 50, 60 17. 1, 1, 3, 4, 5, 9

18. 10, 12, 7, 11, 20, 7, 6, 8, 9 19. 1202, 1229, 1012, 1014, 1120, 1429

20. 3.1, 2.7, 6.0, 5.6, 2.3, 2.0, 1.3 21. 19.4, 16.3, 12.7, 24.8, 19.2, 15.4

BOX-AND-WHISKER PLOTS Draw a box-and-whisker plot of the data set.

22. 1, 2, 2, 4, 5, 7 23. 19, 19, 19, 89, 93, 95

24. 47, 88, 89, 61, 70, 71, 79 25. 40, 100, 20, 40, 100, 70, 90

26. 1.7, 8.5, 1.2, 3.8, 8.5, 5.2, 6.9 27. 61.2, 23.0, 72.7, 74.3, 19.1, 6.6, 28.4


102 4 6 8102 4 6 8

GUIDED PRACTICE

STUDENT HELP


STUDENT HELP


33, 36, 39Examples 2, 3: Exs. 16–21,

34, 37Example 4: Exs. 22–27,

40, 43Examples 5, 6: Exs. 28–32,

41, 44


Skill Check ✓


HISTOGRAMS Use the given intervals to make a frequency distribution of thedata set. Then draw a histogram of the data set.

28. Use five intervals beginning with 1–2.1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9

29. Use five intervals beginning with 10–19.10, 12, 14, 15, 15, 23, 26, 26, 27, 28, 37, 37, 37, 37, 38, 39, 39, 40, 48, 58

30. Use ten intervals beginning with 0–9.66, 9, 43, 28, 5, 7, 90, 9, 78, 6, 69, 43, 28, 55, 13, 64, 45, 10, 54, 96

31. Use fifteen intervals beginning with 0.0–0.4.2.2, 5.6, 2.2, 4.4, 2.2, 6.6, 2.4, 5.8, 2.4, 4.8, 2.4, 6.1, 5.4, 5.4

32. Use eight intervals beginning with 15.5–20.4.22.2, 22.2, 22.6, 24.3, 24.3, 24.8, 54.5, 54.4, 54.1, 52.3, 52.3, 52.9

SCRATCH PROTECTION In Exercises 33–35, use the following information. Computer chips have a silicon dioxide coating that gives them scratch protection.Phosphorus is contained in this coating and is a key to its effectiveness. A companythat produces silicon dioxide has two machines that vary from run to run on thephosphorus content. The phosphorus contents for four runs on each machine areshown below.

33. Find the mean, median, and mode of each data set.

34. Find the range and standard deviation of each data set.

35. CRITICAL THINKING Based on the data, which machine is more consistent?Explain your reasoning.

REAL ESTATE In Exercises 36–38, suppose a real estate agent has soldseven homes priced at $104,900, $119,900, $134,900, $142,000, $179,900,$199,900, and $750,000.

36. Find the mean, median, and mode of the selling prices.

37. Find the range and standard deviation of the selling prices.

38. CRITICAL THINKING How does the selling price $750,000 affect the mean?How does this explain why the most commonly used measure of centraltendency for housing prices is the median rather than the mean or the mode?

39. POLITICAL SURVEY A random survey of 1000 people asked their opinionson a controversial issue. The opinions are categorized as 0 = a negative opinion,1 = a positive opinion, and 2 = a neutral opinion. The results of the survey are 212 category 0 responses, 627 category 1 responses, and 161 category 2responses. Which measure of central tendency is most appropriate to represent the data? Explain.


Machine 1

Run Percent of coating thatnumber is phosphorous (by weight)

1 2.63

2 2.72

3 2.47

4 2.55

Machine 2

Run Percent of coating thatnumber is phosphorous (by weight)

1 1.09

2 3.06

3 4.10

4 2.12

CHEMICALENGINEER

Chemical engineers work in a variety of fields such aschemical manufacturing,electronics, and photographicequipment. Some testmethods of production, asdiscussed in Exs. 33–35.


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In Exercises 40–42, use the tables below which give theages of the Presidents and Vice Presidents of the United States.

40. Draw a box-and-whisker plot of each data set.

41. Make a frequency distribution of each data set using five intervals beginning with30–39. Then draw a histogram of each data set.

42. VISUAL THINKING What is one conclusion you can draw about the ages of thepresidents and vice presidents based on your graphs?

In Exercises 43–45, use the tables below which givethe margins of victory for each championship game in the AFC and in the NFCfor the 1966–1998 seasons.


44. Make a frequency distribution of each data set using five intervals beginning with 1–10. Then draw a histogram of each data set.

45. VISUAL THINKING What is one conclusion you cannot draw about the marginsof victory in the AFC and NFC championship games based on your graphs?

46. RESEARCH Research two sets of data that you can compare. Find the measuresof central tendency and measures of dispersion of each data set. Then draw abox-and-whisker plot and a histogram of each data set. What do you observe?

47. MULTIPLE CHOICE What is the mean of 2, 2, 6, 7, 9, 10?

¡A 2 ¡B 6 ¡C 6.5 ¡D 7 ¡E 7.2

48. MULTIPLE CHOICE What is the median of 0.5, 0.6, 0.7, 1.2, 1.5, 1.5?

¡A 0.7 ¡B 0.95 ¡C 1 ¡D 1.2 ¡E 1.5

49. MULTIPLE CHOICE What is (are) the mode(s) of 12, 13, 13, 15, 16, 16?

¡A 13 ¡B 14 ¡C 13, 16 ¡D 17 ¡E none

50. ALTERNATE FORMULA The formula for standard deviation can also be writtenas follows:

� = �� º� xÆ�2�For n = 3, show that this formula is equivalent to the formula given on page 446. (Hint: You will need to show that x1 + x2 + x3 = 3xÆ.)

x12 + x2

2 + . . . + xn2

��n

STATISTICS CONNECTION

HISTORY CONNECTION


Ages of the first 42 Presidents of the United States when they first took office

42, 43, 46, 46, 47, 48, 49, 49, 50, 51, 51,51, 51, 51, 52, 52, 54, 54, 54, 54, 55, 55,55, 55, 56, 56, 56, 57, 57, 57, 57, 58, 60,61, 61, 61, 62, 64, 64, 65, 68, 69

Ages of the first 47 Vice Presidents of the United States when they first took office

36, 40, 41, 42, 42, 42, 44, 45, 45, 46, 48,49, 49, 50, 50, 51, 51, 51, 52, 52, 52, 52,52, 53, 53, 53, 53, 56, 56, 56, 57, 57, 58,59, 60, 60, 61, 64, 64, 65, 65, 66, 66, 68,69, 69, 71

TestPreparation

★★ Challenge

AFC Championship margins of victory

24, 33, 4, 10, 10, 21, 4, 17, 11, 6, 17, 3,29, 14, 7, 20, 14, 16, 17, 17, 3, 5, 11, 16,48, 3, 19, 17, 4, 4, 14, 3, 13

� Source: Facts About the Presidents

NFC Championship margins of victory

7, 4, 34, 20, 7, 11, 23, 17, 4, 30, 11, 17,28, 9, 13, 1, 14, 3, 23, 24, 17, 7, 25, 27,2, 31, 10, 17, 10, 11, 17, 13, 3

EXTRA CHALLENGE




EVALUATING EXPRESSIONS Evaluate the expression for the given value of x.(Review 1.2 for 8.1)

51. x5 º 8 when x = 2 52. 3x3 + 7 when x = �73

�

53. (7x)3 + 17 when x = º1 54. �x4

4ºx

1� when x = 0.5

PROPERTIES OF EXPONENTS Evaluate the expression. Tell which propertiesof exponents you used. (Review 6.1 for 8.1)

55. 33 • 34 56. (4º3)2 57. (º2)(º2)º3

58. (5º2)º2 59. 10º2 • 100 60. 70 • 72 • 7º2

GRAPHING POLYNOMIALS Graph the polynomial function. (Review 6.2)

61. ƒ(x) = 3x5 62. ƒ(x) = x3 º 4 63. ƒ(x) = ºx4 + 3x2 º 5

64. ƒ(x) = ºx3 + 2x 65. ƒ(x) = x4 º 6x2 º 9 66. ƒ(x) = x5 º 2x + 4

Graph the function. Then state the domain and range. (Lesson 7.5)

1. y = �x�+� 8� 2. y = (x + 7)1/2 º 2 3. y = 3�3 x�º� 6�

Solve the equation. Check for extraneous solutions. (Lesson 7.6)

4. �4 2�x� = 5 5. �3�x�+� 7� = x º 1 6. �3 2�x� º 2�3 x� = 0

Find the mean, median, mode, range, and standard deviation of the data set.(Lesson 7.7)

7. 0, 1, 2, 2, 5, 6, 6, 6, 7, 9 8. 15, 32, 18, 21, 26, 12, 43

9. The radius r of a sphere is related to the volume V of

the sphere by the formula r = 0.620�3 V�. Graph the formula. Then estimate thevolume of a sphere with a radius of 10 units. (Lesson 7.5)

10. ASTRONOMY Kepler’s third law of planetary motion states that a planet’sorbital period P (in days) is related to its orbit’s semi-major axis a (in millions ofkilometers) by the formula P = 0.199a3/2. The orbital period of Mars is about1.88 years. What is its semi-major axis? (Lesson 7.6)

In Exercises 11 and 12, use the following data set ofyears when each state was admitted to statehood. (Lesson 7.7)

1819, 1959, 1912, 1836, 1850, 1876, 1788, 1787, 1845, 1788, 1959, 1890, 1818, 1816, 1846, 1861, 1792, 1812, 1820, 1788, 1788, 1837, 1858, 1817, 1821, 1889, 1867, 1864, 1788, 1787, 1912, 1788, 1789, 1889, 1803, 1907, 1859, 1787, 1790, 1788, 1889, 1796, 1845, 1896, 1791, 1788, 1889, 1863, 1848, 1890

11. Draw a box-and-whisker plot of the data set.

12. Make a frequency distribution of the data set using five intervals beginning with1750–1799. Then draw a histogram of the data set.

HISTORY CONNECTION

GEOMETRY CONNECTION

QUIZ 3 Self Test for Lessons 7.5–7.7

MIXED REVIEW


7.7 Technology Activity 453

Using Technology

Graphing Calculator Activity for use with Lesson 7.7ACTIVITY 7.7

Statistics and Statistical GraphsYou can use a graphing calculator to find statistics and draw statistical graphs.

� EXAMPLEThe fat content and number of calories in several different sandwiches available at a restaurant are shown in the tables below. Use a graphing calculator to (a) find the mean, median, range, and standard deviation of the fat content in the sandwiches, (b) draw a box-and-whisker plot of the fat content in the sandwiches, and (c) draw a histogram of the number of calories in the sandwiches.

� SOLUTIONa. Use the Stat Edit feature to enter

the data in a list. Then use the StatCalc menu to choose 1-variable statistics.

b. Use the Stat Plot menu to choose the type of plot (box-and-whisker), the list of data, and the frequency for the data. Then set an appropriate viewing window.

STUDENT HELP

Study TipThe lower quartile is also known as the first quartile (Q1 on agraphing calculator), and the upper quartile is also known as thethird quartile (Q3 on agraphing calculator).

The mean is xÆ and the standarddeviation is �x. By scrolling you canfind the median (Med). The range isthe difference of maxX and minX.

Draw the box-and-whisker plot. Use the Trace feature to view theminimum (9), the lower quartile(20), the median (25), the upperquartile (30), and the maximum (34).

STUDENT HELP

KEYSTROKE HELP

See keystrokes for several models ofcalculators atwww.mcdougallittell.com

INTE

RNET

EDIT CALC1:1-Var Stats2:2-Var Stats3:Med-Med4:LinReg(ax+b)5:QuadReg6 CubicReg

1-Var Stats x=23.6 �x=236 �x2=6142 Sx=7.974960815 �x=7.565712128 n=10

Plot1 Plot2 Plot3On OffType:

XList:L1Freq:1

Med=25

P1

Sandwich Fat (g) Calories

Hamburger 9 260

Cheeseburger 13 320

Quarter-pound 21 420hamburger

Quarter-pound 30 530cheeseburger

Double cheeseburger 31 560


Bacon cheeseburger 34 590

Fried chicken 25 500

Grilled chicken 20 440

Breaded fish on 28 560deluxe roll

Breaded fish on 25 450plain bun



c. Use the Stat Edit feature to enter the data, and use the Stat Plot menu to choose the type of plot (histogram), the list of data, and the frequency for the data. Then set an appropriate viewing window.

� EXERCISES

In Exercises 1–5, use the tables below which give the fat content and the numberof calories in several different sandwiches available at a competing restaurant.

1. Use a graphing calculator to find the mean, median, range, and standarddeviation of the fat content in the sandwiches.

2. How does the mean fat content in the sandwiches from the restaurant abovecompare with the mean fat content in the sandwiches from the restaurant in theexample? Which restaurant’s sandwiches have a higher median fat content?Which restaurant’s sandwiches have a lower standard deviation of fat content?Based on these statistics, write a sentence that compares the fat content in thesandwiches of the two restaurants.

3. Use a graphing calculator to draw a box-and-whisker plot of the fat content in thesandwiches.

4. Use a graphing calculator to draw a box-and-whisker plot of the fat content in thesandwiches from the restaurant in the example in the same viewing window asthe plot from Exercise 3. Which restaurant’s sandwiches have a lower fatcontent? Write a sentence explaining how the box-and-whisker plots help youanalyze the data.

5. Use a graphing calculator to draw a histogram of the number of calories in thesandwiches. Use an interval width of 50 for each. Write a sentence comparingthe number of calories in the sandwiches from the restaurant above with thenumber of calories in the sandwiches from the restaurant in the example on page 453.

STUDENT HELP

Study TipThe interval width for ahistogram is determinedby the x-scale whensetting an appropriateviewing window.

Draw the histogram. Use theTrace feature to view theminimum of each interval, themaximum of each interval, andthe frequency of each interval.

Plot1 Plot2 Plot3On OffType:

XList:L2Freq:1

max<450 n=2

P2

min=400


Plain hamburger 16 360

Hamburger 20 420with everything

Bacon cheeseburger 30 580

Small hamburger 10 270

Small cheeseburger 17 360


Small bacon 19 380cheeseburger

Grilled chicken 8 310

Breaded chicken 18 440

Chicken club 20 470

Spicy chicken 15 410


WHY did you learn it? Find the number of reptile and amphibian species thatPuerto Rico can support. (p. 405)

Model frequencies in the musical range of a trumpet.(p. 413)

Find the height of a dinosaur. (p. 419)

Find your bowling average. (p. 428)

Find the age of an African elephant. (p. 433)

Determine which boats satisfy the rule for competing in the America’s Cup. (p. 443)

Find surface areas of mammals. (p. 410)

Find wind speeds that correspond to Beaufort wind scale numbers. (p. 440)

Analyze data sets such as the free-throw percentagesfor the players in the WNBA. (pp. 445 and 446)

Graph data sets such as the ages of the Presidentsand Vice Presidents of the United States. (p. 451)

Chapter SummaryCHAPTER

7

WHAT did you learn?Evaluate nth roots of real numbers. (7.1)

Use properties of rational exponents to evaluate andsimplify expressions. (7.2)

Perform function operations. (7.3)

Find inverses of linear and nonlinear functions. (7.4)

Graph square root and cube root functions. (7.5)

Solve equations that contain radicals or rational exponents. (7.6)

Use roots and rational exponents in real-life problems. (7.1–7.6)

Use power functions, inverse functions, and radicalfunctions to solve real-life problems. (7.3–7.6)

Use measures of central tendency and measures of dispersion to describe data sets. (7.7)

Represent data graphically with box-and-whisker plots and histograms. (7.7)

How does Chapter 7 fit into the BIGGER PICTURE of algebra?In Chapter 7 you saw the familiar ideas of squares and square roots extended. Thiswas a significant step in your study of powers and roots as you used exponents thatwere not whole numbers in expressions, functions, and many real-life problems. Youwill continue to build on these ideas as long as you study mathematics.

How did you quizyourself?Here is an example of a quiz thatwas written for Lesson 7.3 andused before a class quiz wasgiven, following the StudyStrategy on page 400.

STUDY STRATEGY

Quiz Yourself

Let ƒ(x) = º2x and g(x) = x º 4. Perform the indicated operation and state the domain. (Lesson 7.3)

1. ƒ(x) + g(x) 2. ƒ(x) º g(x)

3. ƒ(x) • g(x) 4. �ƒ

g(

(

x

x

)

)�

5. g( ƒ(x)) 6. ƒ(ƒ(x))

455



Chapter ReviewCHAPTER

7

• n th root of a, p. 401

• index, p. 401

• simplest form, p. 408

• like radicals, p. 408

• power function, p. 415

• composition, p. 416

• inverse relation, p. 422

• inverse function, p. 422

• radical function, p. 431

• extraneous solution, p. 439

• statistics, p. 445

• measure of central tendency, p. 445

• mean, p. 445

• median, p. 445

• mode, p. 445

• measure of dispersion, p. 446

• range, p. 446

• standard deviation, p. 446

• box-and-whisker plot, p. 447

• lower quartile, p. 447

• upper quartile, p. 447

• histogram, p. 448

• frequency, p. 448

• frequency distribution, p. 448

VOCABULARY

7.1 NTH ROOTS AND RATIONAL EXPONENTS

You can evaluate nth roots using radicals or rational exponents.

Radical notation: 27º2/3 = = = �312� = �9

1�

Rational exponent notation: 27º2/3 = �27

12/3� = = �

312� = �

19�

1�(271/3)2

1�(�3 2�7�)2

1�272/3

Examples onpp. 401–403

Evaluate the expression without using a calculator.

1. �4 1�6� 2. (�3 6�4�)2 3. 9º5/2 4. 2161/3 5. �5 º�3�2�

6. Find the real nth root(s) of a if n = 4 and a = 81.

7. Find the real nth root(s) of a if n = 5 and a = º1.

8. Find the real nth root(s) of a if n = 7 and a = 0.

7.2 PROPERTIES OF RATIONAL EXPONENTS

You can use properties of rational exponents to simplify expressions.

�3 1�2� • �3 4� = �3 1�2� •� 4� = �3 4�8� = �3 8� •� 6� = �3 8� • �3 6� = 2�3 6�

= = = x(1 º 1/2)y(2 º 3/4) = x1/2y5/4xy2�x1/2y3/4

x(1/2 • 2)y2�

x1/2y3/4(x1/2y)2�x1/2y3/4



9. 51/4 • 5º9/4 10. (1001/3)3/4 11. �3 �1�10�6

0�0�� 12. 5�3 1�7� º 4�3 1�7�

13. (81x)1/4 14. �(

(

4

4

x

x

)

)1

2

/2� 15. �6 6�x6�y7�z1�0� 16. �3 4�a�6� + a�3 1�0�8�a�3�

EXAMPLES

EXAMPLES


Chapter Review 457

7.3 POWER FUNCTIONS AND FUNCTION OPERATIONS

You can add, subtract, multiply, or divide any two functions ƒ and g. Youcan also find the composition of any two functions.

Let ƒ(x) = 2x1/2 and g(x) = x4

Addition: ƒ(x) + g(x) = 2x1/2 + x4

Multiplication: ƒ(x) • g(x) = 2x1/2 • x4 =2x9/2

Composition: ƒ(g(x)) = ƒ(x4) = 2(x4)1/2 = 2x2


Let ƒ(x) = 2x º 4 and g(x) = x º 2. Perform the indicated operation.

17. ƒ(x) + g(x) 18. ƒ(x) º g(x) 19. ƒ(x) • g(x) 20. 21. ƒ(g(x))ƒ(x)�g(x)

7.4 INVERSE FUNCTIONS

You can find the inverse relation of any function. To verify that twofunctions are inverses of each other, show that ƒ(ƒº1(x)) = ƒº1(ƒ(x)) = x.

ƒ(x) = y = 2x º 5

x = 2y º 5

x + 5 = 2y

�12�x + �

52� = y = ƒº1(x)



22. ƒ(x) = º2x + 1 23. ƒ(x) = ºx4, x ≥ 0 24. ƒ(x) = 5x3 + 7

25. Verify that ƒ(x) = º2x5 and g(x) = �5 º��2x

�� are inverse functions.

7.5 GRAPHING SQUARE ROOT AND CUBE ROOT FUNCTIONS

You can graph a square root function by starting

with the graph of y = �x�. You can graph a cube root function by

starting with the graph of y = �3 x�.

To graph y = �3 x�º� 5� º 2, first sketch y = �3 x� (shown in red).Then shift the graph right 5 units and down 2 units. From thegraph of y = �3 x�º� 5� º 2, you can see that the domain and rangeof the function are both all real numbers.



26. y = (x º 7)1/3 27. y = �x� + 6 28. y = º2(x º 3)1/2 29. y = 3�3 x�+� 4� º 9

EXAMPLE

EXAMPLES

EXAMPLES

ƒ(ƒº1(x)) = 2��12�x + �

52�� º 5 = x + 5 º 5 = x

ƒº1(ƒ(x)) = �12�(2x º 5) + �

52� = x º �

52� + �

52� = x

1

1

y

x

y � �x3

y � �x � 5 � 23



7.6 SOLVING RADICAL EQUATIONSExamples onpp. 437–440

Solve the equation. Check for extraneous solutions.

30. 3(x + 1)1/5 + 5 = 11 31. �3 5�x�+� 3� º �3 4�x� = 0 32. �4�x� = x º 8

7.7 STATISTICS AND STATISTICAL GRAPHS

The table shows the normal daily high temperatures (in degreesFahrenheit) for Phoenix, Arizona, from 1961 to 1990.

MEAN Find the average of the numbers: = �101320.8� = 85.9

MEDIAN Write the numbers in increasing order and locate the middle number(s):65.9, 66.2, 70.7, 74.9, 75.5, 84.5, 88.1, 93.6, 98.3, 103.5, 103.7, 105.9

There are two middle numbers, so find their mean: �84.5 +2

88.1� = 86.3

MODE Find the number(s) that occur most frequently: none

RANGE Find the difference between the greatest and least numbers: 105.9 º 65.9 = 40

STANDARD Use the formula: ��≈ 14.4DEVIATION

BOX-AND- Find the quartiles: �70.7 +2

74.9� = 72.8 and �98.3 +

2103.5� = 100.9

WHISKERPlot the minimum, maximum, median, and quartiles. Then draw the box and thePLOTwhiskers (not shown).

HISTOGRAM Using five intervals beginning with 60–69, tally the data values for each interval.Then draw a histogram of the data set (not shown).

(65.9 º 85.9)2 + (70.7 º 85.9)2 + . . . + (66.2 º 85.9)2

��12

65.9 + 70.7 + . . . + 66.2��12


In Exercises 33 and 34, use the following data set of employees’ ages at a smallcompany: 21, 25, 30, 36, 39, 40, 44, 45, 46, 51, 51, 63.

33. Find the mean, median, mode, range, and standard deviation of the data set.

34. Draw a box-and-whisker plot and a histogram of the data set. For the histogram, use five intervalsbeginning with 20–29.

You can solve equations that contain radicals or rational exponents byraising each side of the equation to the same power.

�x�º� 4� = 6

(�x�º� 4�)2 = 62 Square each side.

x º 4 = 36

x = 40

EXAMPLES

EXAMPLES

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

65.9 70.7 75.5 84.5 93.6 103.5 105.9 103.7 98.3 88.1 74.9 66.2

4x2/3 = 100

x2/3 = 25

(x2/3)3/2 = 253/2 Raise each side to }32} power.

x = 125


Chapter Test 459

Chapter TestCHAPTER

7Evaluate the expression without using a calculator.

1. �3 º�1�0�0�0� 2. 45/2 3. (º64)2/3 4. 243º1/5 5. �4 1�6�


6. (21/3 • 51/2)4 7. �3 2�7�x3�y6�z9� 8. �1

3

2

x

x

y1

º

/2

1

y� 9. ��

81yx2��3/4

10. �1�8� + �2�0�0�

Perform the indicated operation and state the domain.

11. ƒ + g; ƒ(x) = x º 8, g(x) = 3x 12. ƒ º g; ƒ(x) = 2x1/4, g(x) = 5x1/4

13. ƒ • g; ƒ(x) = 5x + 7, g(x) = x º 9 14. �ƒg�; ƒ(x) = xº1/5, g(x) = x3/5

15. ƒ(g(x)); ƒ(x) = 4x2 º 5, g(x) = ºx 16. g(ƒ(x)); ƒ(x) = x2 + 3x, g(x) = 2x + 1


17. ƒ(x) = �13�x º 4 18. ƒ(x) = º5x + 5 19. ƒ(x) = �

34�x2, x ≥ 0 20. ƒ(x) = x5 º 2


21. ƒ(x) = �x�º� 6� 22. ƒ(x) = �3 x� + 3 23. ƒ(x) = 3(x + 4)1/3 º 2 24. ƒ(x) = º2x1/2 + 4

Solve the equation. Check for extraneous solutions.

25. x5/2 º 10 = 22 26. (x + 8)1/4 + 1 = 0 27. �3 7�x�º� 9� + 11 = 14 28. �4�x�+� 1�5� º 3�x� = 0

29. Some biologists study the structure of animals. By studying a series of antelopes, biologists have found that the length l(in millimeters) of an antelope’s bone can be modeled by

l = 24.1d2/3

where d is the midshaft diameter of the bone (in millimeters). If the bone of an antelope has a midshaft diameter of 20 millimeters, what is the length of the bone? � Source: On Size and Life

ACADEMY AWARDS In Exercises 30–33, use the tables below which givethe ages of the Academy Award winners for best actress and for best actorfrom 1980 to 1998.

30. Find the mean, median, mode, range, and standard deviation of each data set.


32. Make a frequency distribution of each data set using six intervals beginning with21–30. Then draw a histogram of each data set.

33. Writing Compare the ages of the best actresses with the ages of the bestactors. Use statistics and statistical graphs to support your statements.

BIOLOGY CONNECTION

Best actress

21, 25, 26, 29, 31, 33, 33, 34, 34, 38, 39, 41, 42, 45, 49, 49, 61, 72, 80

Best actor

30, 32, 35, 37, 37, 38, 39, 42, 43, 45,45, 46, 51, 52, 52, 54, 60, 61, 76



Chapter Standardized Test CHAPTER

7

1. MULTIPLE CHOICE If x4 = 625, what does x equal?

¡A 5 ¡B º5 ¡C ±5

¡D 25 ¡E ±25

2. MULTIPLE CHOICE What is the simplified form of

the expression �1�8� + �2�0�0� + �2� º �8�?

¡A 12�2� ¡B 14�2�

¡C 18�2� ¡D 14�2� º �8�

¡E 4�2� º 4�8�

3. MULTIPLE CHOICE What is the simplified form of

the expression �3 5�4�x3�y6�z1�0�? (Assume all variablesare positive.)

¡A xy2�3 5�4�z1�0� ¡B xy2z3�3 5�4�z�

¡C 3y3z7�3 5� ¡D 3xy2z3�3 2�z�

¡E 18xy3z7

4. MULTIPLE CHOICE Which of the following is trueif ƒ(x) = 3xº1/2, g(x) = 6x3/4, and h(x) = 18x1/4?

¡A h(x) = ƒ(x) + g(x) ¡B h(x) = ƒ(x) º g(x)

¡C h(x) = ƒ(x) • g(x) ¡D h(x) =

¡E h(x) = ƒ(g(x))

5. MULTIPLE CHOICE If ƒ(x) = x2 º 3x + 7 andg(x) = x2 + 2, what is ƒ(g(x))?

¡A x4 + x2 + 17 ¡B x4 + x2 + 5

¡C x4 + x2 º 9 ¡D x4 + x2 º 3

¡E x4 + 7x2 + 5

6. MULTIPLE CHOICE Which function is the inverse

of ƒ(x) = �12�x º 5?

¡A ƒº1(x) = 2x º 10

¡B ƒº1(x) = 2x + 5

¡C ƒº1(x) = 2x + 10

¡D ƒº1(x) = �12�x + 5

¡E ƒº1(x) = �12�x + �2

5�

7. MULTIPLE CHOICE Which function is graphed?

¡A y = �3 x�º� 3� + 8 ¡B y = �3 x�+� 3� + 8

¡C y = �3 x�º� 8� + 3 ¡D y = �3 x�+� 8� º 3

¡E y = �3 x�+� 8� + 3

8. MULTIPLE CHOICE What is the solution of theequation (3x + 5)1/2 º 3 = 4?

¡A º�43� ¡B �

83� ¡C �

131�

¡D �134� ¡E �

434�

9. MULTIPLE CHOICE What is the solution of the

equation 4�3 x�º� 5� = 20?

¡A 120 ¡B 130 ¡C 220

¡D 2005 ¡E 4101

10. MULTIPLE CHOICE What is the median of6, 4, 4, 10, 5, 12, 1?

¡A 4 ¡B 5 ¡C 6

¡D 7 ¡E 10

11. MULTIPLE CHOICE Which data set matches thebox-and-whisker plot shown?

¡A 1, 1, 3, 5, 6, 7, 9 ¡B 1, 2, 3, 5, 7, 8, 9

¡C 1, 2, 4, 5, 6, 8, 9 ¡D 1, 3, 4, 5, 6, 7, 9

¡E 1, 3, 5, 5, 7, 8, 9

97531

0 109876554321

1

2

y

x

ƒ(x)�g(x)

TEST-TAKING STRATEGY Some college entrance exams allow the optional use of calculators.If you do use a calculator, make sure it is one you are familiar with and have used before.


QUANTITATIVE COMPARISON In Exercises 12 and 13, choose the statementthat is true about the given quantities.





12.

13.

14. MULTI-STEP PROBLEM The metabolic rate r (in kilocalories per day) of amammal can be modeled by r = km3/4 where k is a constant and m is the mass (in kilograms) of the mammal. The specific metabolic rate s (the rate per unit mass)

can be modeled by s = �kmm

3/4�. � Source: Scaling: Why is Animal Size so Important?

a. A 922 kilogram cow has a metabolic rate of about 11,700 kilocalories per day.What is the value of k in the model for metabolic rate?

b. Using the k-value from part (a), simplify the model given for specificmetabolic rate.

c. What is the specific metabolic rate of a 922 kilogram cow?

d. What is the specific metabolic rate of a 16 gram mouse?

e. Writing How does the specific metabolic rate change with decreasingbody mass?

15. MULTI-STEP PROBLEM Follow the steps below to find the relationshipbetween the number of pedal revolutions of a bicycle and the distance traveled.

a. The rear wheel of a bicycle has a diameter of 70 centimeters. Write thefunction that describes the distance d traveled by the bicycle in terms of thenumber w of rear-wheel revolutions. (Hint: When w = 1, the distance traveledby the bicycle is equal to the circumference of the rear wheel.)

b. The gear ratio of a bicycle is calculated by dividing the number of teeth in the chainwheel by the number of teeth in the freewheel. The number w of rear-wheel revolutions is equal to the product of the gear ratio and the number p ofpedal revolutions. A bicycle in first gear has 24 teeth in the chainwheel and 32teeth in the freewheel. Write the function that describes w in terms of p.

c. Use composition of functions to find the relationship between d and p.

d. Shifting gears on a bicycle changes the gear ratio. Use the table below to findhow the distance traveled per pedal revolution changes as you shift gears.

Chapter Standardized Test 461

Gear Number of teeth in chainwheel Number of teeth in freewheel

5th 24 19

10th 40 22

15th 50 19

Column A Column B

ƒ(8) where ƒ(x) = xº2/3 ƒ(2) where ƒ(x) = xº2

ƒ(ƒ(0)) where ƒ(x) = 5x º 2 ƒ(ƒ(0)) where ƒ(x) = x3 + 1


POWERS,ROOTS, AND RADICALS

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