Conjugates are expressions, like 1a + 1b and 1a - 1b, that differ only in the signs
of the second terms. When a and b are rational numbers, the product of two radical
conjugates is a rational number.
Problem 5 Multiplying Conjugates
Where have you seen #What is the product (5 − 17)(5 + 17)? Distribute.
conjugates before? (5 - 17)(5 + 17) = 5 5 + 517 - 517 - ( 17)2 Simplify.
= 25 - 7
The complex number
a + bi has a conjugate,
a - bi. Multiplying
them results in a number = 18
with no imaginary part. Got It? 5. What is each product?
a. (6 - 112)(6 + 112) b. (3 + 18) (3 - 18)
Sometimes a denominator is a sum or difference involving radicals. If the radical
expressions are square roots, you can rationalize the denominator by multiplying the
numerator and the denominator by the conjugate of the denominator.
Problem 6 Rationalizing the Denominator
How can you write the expression with a rationalized denominator?
3 12
#15 − 12
What is a rationalized 3 12 = 3 12 15 + 1122 Multiply. Use the conjugate of the denominator.
denominator? 15 - 12 15 - 12 15 +
A rationalized
denominator contains no = 3 12 ( 15 + 12)
radicals. (15)2 - (12)2
# # The radicals in the denominator cancel out.
= 3( 12 15 + 12 12) Distribute 12 in the numerator.
5-2
= 3(110 + 2) Simplify.
3
= 110 + 2
Got It? 6. How can you write the expression with a rationalized denominator?
a. 132 1- 715 b. 3 4x
- 16
c. Reasoning Suppose you were going to rationalize the denominator of
y12o--u1r1a88n. sWwoeur.ld you simplify 18 before or after rationalizing? Explain
Lesson 6-3 Binomial Radical Expressions 377
Lesson Check Do you UNDERSTAND? MATHEMATICAL
Do you know HOW? PRACTICES
Simplify if possible.
1. 1016 + 216 8. Vocabulary Determine whether each of the
3. 813x - 513x
2. 3 12 + 4 13 2 following is a pair of like radicals. If so, add them.
4. 513 + 112 a. 3x111 and 3x110
b. 213xy and 713xy
Multiply. c. 12113y and 1216y
5. (4 + 13)(4 - 13)
6. (5 + 215)(7 + 415) 9. Compare and Contrast How are the processes of
7. (2 + 312)(1 - 312) multiplying radical expressions and multiplying
polynomial expressions alike? How are the processes
different?
Practice and Problem-Solving Exercises MATHEMATICAL
A Practice Simplify if possible. PRACTICES
10. 516 + 16 11. 613 3 - 213 3 See Problem 1.
14. 141x + 31y 12. 4 13 + 413 3
13. 31x - 51x 15. 7 23 x2 - 2 23 x2
16. The design of a garden path uses stone See Problem 2.
pieces shaped as squares with a side length
of 15 in. Find the length of the path.
Simplify. See Problem 3.
17. 6118 + 3150 18. 14120 - 31125 19. 118 + 132
20. 13 54 + 13 16 21. 313 81 - 213 54 22. 14 32 + 14 48
Multiply. 24. (2 + 17)(1 + 317) See Problem 4.
27. ( 113 + 6)2
23. (3 + 15)(1 + 15) 25. (3 - 412)(5 - 612)
26. ( 13 + 15)2 28. (2 15 + 3 12)2
Multiply each pair of conjugates. See Problem 5.
29. (5 - 111)(5 + 111)
31. (216 + 8)(2 16 - 8) 30. (4 - 213)(4 + 213)
32. ( 13 + 15)( 13 - 15)
Rationalize each denominator. Simplify your answer. See Problem 6.
33. 1 +413 34. 4 - 2 35. 5 + 1133 36. 23 + 18
313 2 - - 218
378 Chapter 6 Radical Functions and Rational Exponents
B Apply 37. Think About a Plan The design on a parquet floor, shown at the right,
is made of equilateral triangles. The side of a large triangle is 6 in.,
and the side of a small triangle is 3 in. Find the total area of the design to
the nearest tenth of a square inch.
• How many large and how many small triangles form the design?
• Can you express the area of an equilateral triangle through its side?
Simplify. 39. 175 + 2148 - 513
41. 175 - 4118 + 2132
38. 172 + 132 + 118 43. 3 13 16 - 4 13 54 + 13 128
40. 5132x + 4198x 45. ( 13 - 17)( 13 + 217)
42. 4 2216y 2 + 3 254y 2 47. ( 112 + 172)2
44. (1 + 172)(5 + 12) 49. ( 1a + 1 + 1a - 1)( 1a + 1 - 1a - 1)
46. ( 1y + 12)( 1y - 712)
48. ( 11.25 - 11.8)( 15 + 10.2)
50. Error Analysis Describe and correct the error made while
3 + 1122.
simplifying the expression 3 - 3+ 2 = 3+ 2 ؒ3+ 2
3- 2 3- 2 3+ 2
STEM 51. Chemistry A scientist found that x grams of Metal A is
32 + ( 2)2 9 + 2 11
completely oxidized in 2x 13 seconds and x grams of = 32 – ( 2 )2 = 9 – 2 = 7
Metal B is completely oxidized in 6x 13 seconds. How much
faster is Metal A oxidized than Metal B?
52. Reasoning Describe the possible values of a such that 172 + 1a simplifies
to a single term.
53. Writing D iscuss the advantages and disadvantages of first simplifying
172 + 132 + 118 in order to estimate its decimal value.
54. Geometry Show that a right triangle with legs of lengths 12 - 1 and 12 + 1 is
similar to a right triangle with legs of lengths 6 - 132 and 2.
55. Open-Ended F ind two pairs of conjugates with a product of 3.
Rationalize the denominators and simplify.
56. 24 -+ 3112277 57. 142++1163 58. 15 3- 121
- 17
59. 121144+x23 60. 22 + 26 61. 127 - 15
21.5 + 20.5 115 - 3
62. 4 +13 213 2 63. 5 + 14 x 64. 4 - 2 13 6
14 x 13 4
Lesson 6-3 Binomial Radical Expressions 379
C Challenge Add or subtract. 66. 15 4 13 - 15 4 13
- +
65. 1 -115 + 1 +115
67. For what values of a and b does 1a + 1b = 1a + b?
68. In the expression 2n xm, m and n are positive integers and x is a real number. The
expression can be simplified.
a. If x 7 0, what are the possible values for m and n?
b. If x 6 0, what are the possible values for m and n?
c. If x 6 0, and an absolute value symbol is needed in the simplified expression,
what are the possible values of m and n?
Standardized Test Prep
S AT/ACT 69. What is the value of the expression (5 - 213)(5 + 213)?
70. What is the value of z in the solution of the system of equations below?
2x - 3y + z = 6
• -x + y - 2z = -5
3x - y - 3z = -7
71. What is the y-value of the y-intercept of the line 5x - 7y = -15?
72. What is the slope of a line perpendicular to the line 2x + 5y = 10?
73. What is the value of p for which the equation x 2 - 12x + 4p = 0 has exactly one
real root?
Mixed Review
Simplify each expression. Rationalize all denominators. See Lesson 6-2.
# 74. 13 3 13 18
53 4 132 1216
0.5x 12 16
75. 76. #77.
# # # 78. 23 2x2 13 4x 79. 17x 214x3 81. 29x 2 225x 2
80. 13x 15x
Find the real and imaginary solutions of each equation. See Lesson 5-3.
84. 125x 3 - 1 = 0
82. 2x 3 - 16 = 0 83. x 3 + 1000 = 0 87. 81x 4 - 1 = 0
85. x 4 - 14x 2 + 49 = 0 86. 25x 4 - 40x 2 + 16 = 0
Get Ready! To prepare for Lesson 6-4, do Exercises 88–91.
Simplify. See p. 978.
91. (3-2)(35)
88. (x 2)3 89. (pq)5 90. (24)(25)
380 Chapter 6 Radical Functions and Rational Exponents
6-4 Rational Exponents CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
NM-ARFNS.A.9.122 .RNe-wRrNite.1e.x2p rReesswiorintse ienxvporlevsinsgiornasdincavloslvainndg
ratdioicnaalsl eaxnpdornaetniotsnaulsienxgpothneenptrsopuesritniegsthofeepxrpoopneertnitess. of
Aexlpsonreenvtise. AwlssoN-MRNAF.AS..1912.N-RN.1.1
MP 1, MP 3, MP 4
Objective To simplify expressions with rational exponents
This is a 1-square. 1
Square
The first diagram shows how to cut two 2
3
It is easy to cut linked 1-squares into congruent pieces, each
with size 23. Show how to cut the three linked 2
one 1-square into 1-squares into congruent pieces, each with 3
congruent pieces size 43. Explain why you 2
awbiothutssizizee2151. ? congruent pieces, each 3
each cannot cut them into
How with size 54.
MATHEMATICAL
PRACTICES
Lesson # # #If ax = 24 a3, then by definition, ax ax ax ax = a3. By adding exponents, a4x = a3,
then 4x = 3e.xSaomxpilse,342.4Tah3is=suag43g. ests an alternative notation for radical expressions in
Vocabulary which, for
• rational
exponent
Essential Understanding You can write a radical expression in an equivalent
form using a fractional (rational) exponent instead of a radical sign.
In general, 1n x = xn1 for any positive integer n. Like the radical form, the exponent form
indicates the principal root.
1 13 64 = 1 14 16 = 1
136 = 362 643 164
Problem 1 Simplifying Expressions with Rational Exponents
What is the simplified form of each expression?
What does the
#d enominator of the
1 11 You can also solve
A 2163 B 72 72
# #frre apcrteiosennatl?exponent 21613 = 13 216 Rewrite as radicals. this problem by adding
11 17
7 the exponents.
72 72 = 17
#Tfrh aectdioennoismtihneationrdeox f tohfe = 23 63
= 17 1 и 1 ϭ 7 1 ϩ 1 ϭ 71ϭ 7
2 2
72 72
the rad ical. = 6 = 272
= 7
Lesson 6-4 Rational Exponents 381
Problem 1
# C 54 1254
11
1 1 = 14 5 14 125 Rewrite as radicals.
# # 54 1254
# = 14 5 125 Property for multiplying radical expressions
= 14 625
Multiply.
= 24 54 Rewrite the radicand.
= 5 Simplify.
Got It? 1. What is the simplified form of each expression? #1 1
1 11 c. 32 122
# a. 642 b. 112 112
If 1n x = 1 it follows from the Laws of Exponents that for all real numbers
m
xn ,
( )2n xm . This leads to the definition of a rational exponent.
= (x m)n1 = 1 1 2 m = 1n x
xn
Key Concept Rational Exponent
If the nth root of a is a real number, m is an integer, and m is in lowest terms, then
1 m n
an = 1n a and an = 2n am = ( 1n a)m. If m is negative, a ≠ 0.
Problem 2 Converting Between Exponential and Radical Forms
3 A What are 3 and y−3.5 in radical form?
7
x7
Does the fraction 3 = 27 x3 or 117 x23
first need to be
x7
simplified? y-3.5 = y-72
No. The fraction is 1
already in lowest terms. = 7
y2
= 1 = 1 = 1 or 1y
1y7 1y 6y y31y y4
B What are 2a5 and 1 15 b23 in exponential form?
2a5 = (a5)21 = 5
a2
1 15 b23 = 1 1 2 3 = 3
b5 b5
Got It? 2. a. What are the expressions w - 5 and w0.2 in radical form?
8
b. What are the expressions 24 x3 and 115 y24 in exponential form?
c. Reasoning Refer to the definition of rational exponent. Explain the need
for the restriction that a ≠ 0 if m is negative.
382 Chapter 6 Radical Functions and Rational Exponents
Problem 3 Using Rational Exponents STEM
Planetary Motion K epler’s Third Law
of Orbital Motion shows how you can
approximate the period P (in Earth
years) it takes a planet to complete
one orbit of the sun. Use the function
3
P = d where d is the distance from
2,
the planet to the sun in astronomical Sun
units (AU—about 93,000,000 miles or Venus Mercury
the distance from Earth to the sun). Earth
How can you find How many Earth years does it take
a 3 power on a Mars to orbit the sun? Mars
2 3
P = d2 Write the formula.
calculator? 3
You can use = (1.52)2 Substitute for d.
(3 2 ) . ≈ 1.87 Use a calculator.
You can also cube the
number and then take It takes Mars approximately 1.87 Earth
the square root, or take years to orbit the sun.
the square root then
cube.
Got It? 3. Find the approximate length (in Earth years)
of each planet’s year.
a. A Venusian year if Venus is 0.72 AU from the sun
b. A Jovian year if Jupiter is 5.46 AU from the sun
All the properties of integer exponents apply to rational exponents.
Properties Properties of Rational Exponents
Let m and n represent rational numbers. Assume that no denominator equals 0.
Property Example Property Example
#1
#am an = am+n 2 = 81 + 2 = 81 = 8 a-m = 1 9-12 = 1 = 1
83 3 3 am 3
83 1
92
#1 3
(am)n = amn 1 2 4 = 52 4 = 52 = 25 am = am - n 72 = 73 - 1 = 71 = 7
(ab)m = ambm an 2 2
152 1
72
# # #1 1 1 1 11
3 53 53
1 ( )a m am ( )5 = = 3
(4 5)2 = 42 52 = 2 52 = bm 1
b 27
273
Lesson 6-4 Rational Exponents 383
Recall from Lesson 6-2 that you simplified products or quotients involving radical
expressions only when they had the same index. However, you can combine
radical expressions with different indexes if you convert them to expressions with
rational exponents.
Problem 4 Combining Radical Expressions
What is 24 x3 in simplest form?
18 x2
The radicands are different, 24 x3 = 3
but both are powers of the 28 x2
same variable. Write the x4
expressions using exponents.
2
Use the division property
for exponents. Subtract the x8
exponents.
= 3 - 2
Simplify, and write in either 8
exponential or radical form. x4
= 3 - 1
4
x4
1
= x2 = 1x
Got It? 4. What is each product or quotient in simplest form? c. 17 1 13 72
a. 13 1 14 32 b. 213 xx23
You can simplify a number with a rational exponent by using the properties of
exponents or by converting the expression to a radical expression.
Problem 5 Simplifying Numbers With Rational Exponents
What is each number in simplest form?
A 16−2.5 Method 2
What is the first step? 16-2.5 = 16-25
Rewrite the decimal Method 1
exponent as a fraction in 16-2.5 = 16-52
lowest terms.
# = (24)-52 = 1 5
= 24 -25 = 1116211625
= 2-10 = 415
1 10124 1
= 210 = = 1024
384 Chapter 6 Radical Functions and Rational Exponents
Problem 2
4
Does it matter that B ( − 32)5
the base is negative?
No; because the Method 1 Method 2
denominator of the 4
exponent is odd, there
will be a real root. ( - 32)5 =
4 4 4
5
( )(-32)5
= 1 ( - 2)5 2 25 - 32
# ( )
= ( - 2)5 4 = 25 ( - 2)5 4
5
= ( - 2)4 = ( - 2)4
= 16 = 16
Got It? 5. What is each number in simplest form? c. 9-3.5
a. 32-53 b. 1643
To write an expression with rational exponents in simplest form, write every exponent
as a positive number.
Problem 6 Writing Expressions in Simplest Form
What is each expression in simplest form?
2
( ) ( ) A −8x1xy 3
B 16y−8 −43
( ) # # # #s tep in simplifying -8x1xy 3 = (-8)3 x3 ((xy)2)3
What is the first 2 22 12 (16y-8)-43 = 16-43 y-8 - 3
4
# # #uasriandgictahleepxrpo rpeesrstioiens = (( - 2)3)23
2 1 (24)-43 y6
x3 (xy)3 =
# # #oRefwerxitpeotnheenratds i?cals = ( -2)2 x
2 1 1 2-3y 6
3
x3 y3 =
using rational e xponents. = 4xy31, or 4x 13 y = y86
Got It? 6. What is each expression in simplest form?
( ) a. (8x15)-13 b. 3
9x 14 y 2
Lesson Check Do you UNDERSTAND? MATHEMATICAL
Do you know HOW? PRACTICES
Simplify each expression.
#1 1 7. Open-Ended Find a nonzero number q such that
1 1
2. 52 52 q(1 - 22) is a rational number. Explain.
1. 1253
3. 25-23 4. 4-3.5 8. Error Analysis E xplain why this 1
simplification is incorrect.
5. 111114 112 6. 13 x 5(4 – 52)
16 x5 9. Reasoning Explain why
11 1
( - 64)3 = - 643 but
11 5(4) – 5(52)
( - 64)2 ≠ - 642.
1
20 – 252
15
Lesson 6-4 Rational Exponents 385
Practice and Problem-Solving Exercises MATHEMATICAL
PRACTICES
A Practice Simplify each expression. See Problem 1.
1 1 1
11. 273
10. 362 1 12. 492
11
11 ( - 3)3 11
# #14. (-3)3 (-3)3
# 13. 102 102 11 #15. 72 212
11 11
#17. 33 93
# 16. 22 322 #18. 34 274
Write each expression in radical form. See Problem 2.
1 1 2 2
19. x6 20. x5 21. x7 22. y5
25. x1.5 26. y1.2
23. y-98 24. t-34
Write each expression in exponential form. 29. 2(7x)3 30. 1 17x23
33. 24 c2
27. 1-10 28. 27x3 34. 23 (5xy)6
31. 23 a2 32. 1 13 a22
Optimal Height The optimal height h of the letters of a message printed on See Problem 3.
2.27
pavement is given by the formula h = 0.00252d Here d is the distance of the
e .
driver from the letters and e is the height of the driver’s eye above the pavement.
All of the distances are in meters. Find h for the given values of d and e.
35. d = 100 m, e = 1.2 m 36. d = 50 m, e = 1.2 m
37. d = 50 m, e = 2.3 m 38. d = 25 m, e = 2.3 m
Find each product or quotient. 29 y3 #41. 15 15 5 # See Problem 4.
13 y9
39. 114 62113 62 40. 42. 17 7 13 7
46. 1x4y
43. 2136 44 #44. 14 18 112 45. 16
13 36 14 x2y8
Simplify each number. 22 2 See Problem 5.
2 48. 643 643 49. ( - 8)3 6
52. 41.5 53. 161.5
47. 83 50. ( - 32)5
51. (32)-54
54. 10,0000.75
Write each expression in simplest form. See Problem 6.
( ) 55. x23 -3 ( )56. x-47 7 ( )57. 2 -1 ( )58. 5 2 -1
3x3 x3
59. 1 - 27x-9213 60. 1 - 32y15 2 1 ( )61. 1 y-23 -6 ( )62. 2 y-61 -12
5
x2 x3
( ) 63. xx-31 -41 ( )64. 1 12 x-23 15
x4 y-31
x2 1 65. ° y-43 ¢ 66. ° ¢
x-11
3
386 Chapter 6 Radical Functions and Rational Exponents
B Apply 67. Think About a Plan T he ratio R of radioactive carbon to nonradioactive carbon
left in a sample of an organism that died T years ago can be approximated by the
formula R = A(2.7)-80T33. Here A is the ratio of radioactive carbon to nonradioactive
carbon in the living organism. What percent of A is left after 2000 years? After 4000
years? After 8000 years?
• What are the known and unknown values?
• How can you use the properties of exponents to solve this problem?
68. The expression 0.036m43 is used in the study of fluids. Which best represents the
value of the expression for m = 46 * 104?
636 1460 1660 16,600
Simplify each number. 1 71. 321.2
74. 1004.5
1 70. ( - 243)5
3
69. ( - 343)3 73. 643.5
72. 2431.2 77. 252
4
75. - ( - 27)-43 10003
76 .
3
1002
STEM 78. Science A desktop world globe has a volume of about 1386 cubic inches. The
radius of Earth is approximately equal to the radius of the globe raised to the
4 3
10th power. Find the radius of Earth. (Hint: Use the formula V = 3 pr for the
volume of a sphere.)
# Simplify each expression. #1 3 33
23
80. y2 y10 81. x5 , x10
79. x7 x14
83. 2 y-41 84. 1 y-31
53 y-12
x3 x2
82. y7 , y14
1 31
1 x4 y2
x2
85. a1861xy1184 b 2
81y16 1 1
16x12
2 8x6 3
27y9
86. a b 87. a b
1
( ) 88. Open-Ended Find three nonzero numbers a such that a 4 + 52 is a rational
number. Can a itself be a rational number? Explain.
89. a. Reasoning S how that 24 x2 = 1x by using the definition of fourth root.
b. Reasoning Show that 24 x2 = 1x by rewriting 24 x2 in exponential form.
#1 1
90. Simplify 42 42 using the following methods. Show all your work.
a. Use the properties of exponents.
b. Simplify each term in the product, then multiply.
c. Convert to radical form, then simplify.
Lesson 6-4 Rational Exponents 387
C Challenge You can define the rules for irrational exponents so that they have the same
properties as rational exponents. Use those properties to simplify each
expression. 92. 33+ 15 93. x4p
31 + 15 x2p
( ) 91. 712 12
1 ( )96. 32+12 2-12
# 94. 5213 25-13
95. 912
STEM 97. Weather Using data for the effect of temperature and wind on an exposed face, the
National Weather Service uses the following formula to determine wind chill.
Wind Chill Index = 35.74 + 0.6215T - 35.75V 0.16 + 0.4275TV 0.16
T is the temperature in degrees Fahrenheit and V is the velocity of the wind in
miles per hour. Frostbite occurs in about 15 minutes when the wind chill index
is -20. Find the wind velocity that produces a wind chill index of -20 when the
temperature is 5°F.
Standardized Test Prep
S AT/ACT 98. What is the simplified value of 1 1 2 -16?
64
99. What positive value of b makes 9x2 - bx + 4 a perfect square trinomial?
100. How many real roots does the cubic polynomial equation x3 - 7x2 + 13x - 4 = 0
have?
101. What is the y-value of the y-intercept of the graph of f (x) = 4 0 x - 2 0 -5?
PERFORMANCE TASK
Apply What You’ve Learned MATHEMATICAL
Look back at the rule for America’s Cup yachts given on page 359. PRACTICES
MP 2
a. Write the expression on the left side of the rule in radical form.
b. What is the unit of measure associated with each variable in the expression? What
is the unit associated with the expression as a whole? Explain.
c. Estimate the value of the expression for the AC45 Wingsail Catamaran detailed on
page 359. Show your work and explain how you chose values for the length, sail
area, and displacement to make the calculation easier.
388 Chapter 6 Radical Functions and Rational Exponents
6 Mid-Chapter Quiz M athX
OLMathXL® for School
R SCHO Go to PowerAlgebra.com
L®
FO
Do you know HOW? Write each expression in radical form.
find all the real square roots of each number.
3 20. y-34
19. m7
1. 100 2. 0.49 Simplify each expression.
Simplify each radical expression. Use absolute value 2 3
symbols when needed.
21. ( - 27)3 22. (16)4
3. 236x2 4. 23 0.008y 3x 6 Write each expression in simplest form.
23. 713 2x - 313 2x
Simplify. 6. 24 32m7n9 24. 2 232x2 + 3 272x2
5. 250x4y8
25. 23 125x6 - 23 27x6
Multiply and simplify. 26. 14 7 - 13 7
# 7. 624x2 229x2y2 27. 1 1y - 1321 1y + 2132
# 8. 13 9 13 9 -4
# 9. 24 16x8 24 x14 ( ) 28.
13
16x4y4
19
x3
29. ° ¢
2
Divide and simplify. y3
10. 236x4 11. 23 64x9y3 30. axx-510 b 2
19x6 13 8x3 5
Simplify. Rationalize all denominators. 31. The radius of a circle can be expressed as r = A inches
12. 1013 81 - 813 24 5p
where r is the radius and A is the area of the circle. If the
area of a circle is 169p in.2, what is its radius?
13. 4 + 112 Do you UNDERSTAND?
4 - 112 32. What are the real roots of 1-16? Explain.
14. 148 - 3127 + 2175 33. Error Analysis Identify the error in this statement.
15. 13 + 163211 + 172 # 1133 yx 13 y 13 xy
13 y y
16. 1x =
16y3
2 0-32
34. Reasoning If = 0, why is undefined?
03
Write each expression in exponential form. 35. Given that x and y are integers, explain why the
product of x + 1y and its conjugate will always be
17. - 117 18. 23 y8 an integer.
11
36. Reasoning E xplain why ( - 8)2 ≠ - (8)2, but
11
( - 27)3 = - (27)3.
Chapter 6 Mid-Chapter Quiz 389
6-5 Solving Square Root and MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Other Radical Equations
MA-ARFESI..A9.122 .SAo-lRveEIs.i1m.2p leSroalvtieonsiaml panledrraatdioicnaall and
eraqduiacatiloenqsuiantoionnesvinaroianbelev,aarniadb.le.,.asnhdow. . h. oshwow
ehxotwraenxetoruans esoulustisoonlustmioanys amriasye.arAisles.oAAls-oCED.A.4
MMPAF1S,.M91P2.2A, -MCEPD3.,1M.4P 4
MP 1, MP 2, MP 3, MP 4
Objective To solve square root and other radical equations
You are a passenger in the car. You are Cell phone
using a cell phone that connects with the tower
cell phone tower shown. The tower has an 3 miles
effective range of 6 mi. How many miles
do you have to finish your call? Justify p
your answer.
Lesson A radical equation is an equation that has a variable in a radicand or a variable with a
rational exponent. If the radical has index 2, the equation is a square root equation. In
Vocabulary this lesson, assume that all radicals and expressions with rational exponents represent
• radical equation real numbers.
• square root
Essential Understanding Solving a square root equation may require that you
equation square each side of the equation. This can introduce extraneous solutions.
To solve a radical equation, isolate the radical on one side of the equation. Then raise
each side to the power suggested by the index.
Do you need to Problem 1 Solving a Square Root Equation
introduce a t sign
here? What is the solution of 3 + 12x − 3 = 8?
No, when you take the 3 + 12x - 3 = 8
square root of each 12x - 3 = 5 Isolate the radical expression.
side of an equation you ( 12x - 3)2 = 52 Square each side.
do, but here you are 2x - 3 = 25
squaring both sides of 2x = 28 Add 3 to each side.
the equation. x = 14 Divide each side by 2.
390 Chapter 6 Radical Functions and Rational Exponents
Check
3 + 12x - 3 = 8 Write the original equation.
3 + 12(14) - 3 ≟ 8 Substitute 14 for x.
Simplify.
3 + 125 ≟ 8
3 + 5≟8
8=8 ✔
Got It? 1. What is the solution of 14x + 1 - 5 = 0?
the form m = k, raise each equation to the power mn ,
either m n is even, then .
( )To solve equations of xn side of the
How can you get or mn
rid of the rational the reciprocal of mn . If xn m = 0 x 0
exponent?
Raise each side to the Problem 2 Solving Other Radical Equations
reciprocal power.
2
A What is the solution of 3(x + 1)3 = 12?
• The equation Solution of the • I solate the exponential expression.
• The power of the equation • U se the inverse of the power to
exponential expression simplify and solve the equation.
2 Divide each side by 3.
3(x + 1)3 = 12 Raise each side to the 3 power.
2
2
Since the numerator of 2 is even, (x 23 ϭ ͉ x ͉
(x + 1)3 = 4 3
3)2
23 3
((x + 1)3)2 = 42
63
(x + 1)6 = 42
0x + 10 = 8
x + 1 = {8
x = 7 or x = -9
The solutions are 7 and -9.
Check 2
2 3(x + 1)3 = 12
3(x + 1)3 = 12
3(7 + 2 ≟ 12 3( -9 + 2 ≟ 12
1)3 1)3
3(23)32 ≟ 12 3(( - 2)3)23 ≟ 12
3(2)2 ≟ 12 3( - 2)2 ≟ 12
12 = 12 ✔ 12 = 12 ✔
Lesson 6-5 Solving Square Root and Other Radical Equations 391
Problem 3
B What is the solution of 325 (x + 1)3 + 1 = 25?
Why is isolating the 3 25 (x + 1)3 + 1 = 25
variable important?
If you raise each side of 3
3 25 (x + 1)3 + 1 = 25
etondthuep53wpiothwaermyooruewill 3(x + 1)5 + 1 = 25 Rewrite the radical using a rational exponent.
complicated equation,
not a simpler one. 3
3(x + 1)5 = 24 Subtract 1 from each side.
3
(x + 1)5 = 8 Divide each side by 3.
35 5 Raise each side to the 5 power.
3
((x + 1)5)3 = 83
x + 1 = 32 Simplify.
x = 31 Subtract 1 from each side.
The solution is 31.
2
Got It? 2. What are the solution(s) of 2(x + 3)3 = 8?
Problem 3 Using Radical Equations STEM
Earth Science F or Meteor Crater in Arizona, the formula d = 2 3V relates the
5 0.3
diameter d of the rim (in meters) to the volume V (in cubic meters). What is the
volume of Meteor Crater? (All values are approximate.)
1.2 km
d = 2 3 V
0.3
5
What is the diameter d = 3V Solve for V. First divide each side by 2.
in meters? 2 5 0.3
1.2 km = 1.2 * 1000 m.
( )d 3 = V Cube each side.
2 0.3
( )0.3d 3 = V Multiply each side by 0.3.
2
( )0.3 1200 3 = V Substitute 1200 for d.
2
64,800,000 = V Simplify.
The volume of Meteor Crater is about 64,800,000 m3.
Got It? 3. Suppose the diameter of a similarly shaped crater is 1 km. What is the
volume of the crater? Use the formula given in Problem 3.
392 Chapter 6 Radical Functions and Rational Exponents
When you raise each side of an equation to a power, it is possible to introduce
extraneous solutions. Therefore, it becomes very important that you check all solutions
in the original equation. A correct solution will give a true statement. An extraneous
solution will give a false statement.
Problem 4 Checking for Extraneous Solutions
What is the solution of 1x + 7 − 5 = x? Check your results.
1x + 7 - 5 = x
How do you square 1x + 7 = x + 5 Isolate the radical.
a binomial?
1 1x + 7 22 = (x + 5)2 Square each side.
Use the formula,
(a + b)2 = x + 7 = x2 + 10x + 25 Simplify.
a2 + 2ab + b2.
0 = x2 + 9x + 18 Combine like terms.
0 = (x + 3)(x + 6) Factor.
x = - 3 or x = - 6 Zero-Product Property
Check 1x + 7 - 5 = x
1-6 + 7 - 5≟ -6
1x + 7 - 5 = x
1-3 + 7 - 5≟ -3 11 - 5 ≟ -6
1 - 5≟ -6
14 - 5 ≟ -3 -4 ≠ -6
2 - 5≟ -3
-3 = -3 ✔ false
The only solution is -3.
Got It? 4. a. What is the solution of 15x - 1 + 3 = x? Check your results.
b. Reasoning When should you check for extraneous solutions? Explain.
In this lesson you studied algebraic methods of solving square root and radical
equations. In Lesson 6-8 you will study the graphs of square root functions. These
graphs can help you find solutions and identify extraneous solutions.
The calculator screen shows the graphs Y1 = 1 (x + 7) − 5 and Y2 = x. From the
graph, it is clear that -3 is a solution of 1x + 7 - 5 = x, and -6 is not a solution.
Lesson 6-5 Solving Square Root and Other Radical Equations 393
If an equation contains two radical expressions (or two terms with rational exponents),
isolate one of the radicals (or one of the terms), then eliminate it (or its rational
exponent). Isolate the more complicated radical expression first. In the resulting
equation, simplify the expressions before you eliminate the second radical.
Problem 5 Solving an Equation With Two Radicals
Which radical What is the solution of 12x + 1 − 1x = 1?
expression should
you isolate first? 12x + 1 - 1x = 1
Isolate the more
complicated radical first, 12x + 1 = 1x + 1 Isolate the more complicated radical.
12x + 1.
1 12x + 122 = 1 1x + 122 Square each side.
2x + 1 = x + 2 1x + 1
x = 2 1x Isolate 21x.
x2 = 12 1x22 Square each side.
x2 = 4x
x2 - 4x = 0 Subtract 4x from each side.
x(x - 4) = 0 Factor.
x = 0 or x = 4 Zero-Product Property
Check 12x + 1 - 1x = 1
12(4) + 1 - 14 ≟ 1
12x + 1 - 1x = 1
12(0) + 1 - 10 ≟ 1 19 - 14 ≟ 1
3 - 2≟1
11 - 0 ≟ 1
1 - 0≟1 1=1 ✔
1 = 1 ✔
The solutions are 0 and 4.
Got It? 5. What is the solution of 15x + 4 - 1x = 4?
Lesson Check
Do you know HOW? Do you UNDERSTAND? MATHEMATICAL
PRACTICES
Solve. Check for extraneous solutions. 7. Vocabulary W hich value, 12 or 3, is an extraneous
solution of x - 6 = 13x? Explain your reasoning.
1. 14x - 23 - 3 = 2 2. - 13 x + 3 = 0
8. Compare and Contrast How is solving a square
3. 5 1x + 7 = 8 4. 3 1x = 6 root equation similar to solving an absolute value
5. 5 - 2 1x = 3 6. 13 x = 8 equation? How is it different?
394 Chapter 6 Radical Functions and Rational Exponents
Practice and Problem-Solving Exercises MATHEMATICAL
PRACTICES
A Practice Solve. See Problem 1.
9. 3 1x + 3 = 15 10. 4 1x - 1 = 3 11. 1x + 3 = 5
12. 1x + 1 = 4 13. 12x - 1 = 3 14. 1x + 2 - 2 = 0
15. 13x + 4 = 4 16. 12x + 3 - 7 = 0 17. 16 - 3x - 2 = 0
Solve. 2 See Problem 2.
2 19. (x + 2)3 = 9 3
18. (x + 5)3 = 4 3 20. 3(x - 2)4 = 24
3 22. (x + 1)2 - 2 = 25 3
21. 3(x + 3)4 = 81 23. 3 + (4 - x)2 = 11
24. Volume A spherical water tank holds 9000 ft3 of water. What is the See Problem 3.
1 d3p V2
diameter of the tank? 1 Hint : 6 =
STEM 25. Hydraulics T he formula pd 2v = Q models the diameter of a pipe where Q is the
4
maximum flow of water in a pipe, and v is the velocity of the water. What is the
diameter of a pipe that allows a maximum flow of 30 ft3>min of water flowing at a
velocity of 400 ft/min? Round your answer to the nearest inch.
Solve. Check for extraneous solutions. See Problem 4.
28. 1-3x - 5 = x + 3
26. 13x + 7 = x - 1 1 31. 13x + 13 - 5 = x
34. 1x + 7 - x = 1
27. (5 - x)2 = x + 1
29. 111x + 3 - 2x = 0 1
30. (5x - 4)2 - x = 0
32. 1x + 7 + 5 = x 1
33. (x + 3)2 - 1 = x
Solve. Check for extraneous solutions. 11 See Problem 5.
35. 13x = 1x + 6 36. (2x)2 = (x + 5)2 s
11 s
38. 13x + 2 - 12x + 7 = 0 s√3
37. (7x + 6)2 - (9 + 4x)2 = 0 2
11
11
40. (x - 2)2 - (28 - 2x)4 = 0
39. (x + 5)2 - (5 - 2x)4 = 0
41. 15 - x - 1x = 1 42. 13x + 1 - 1x + 1 = 2
43. 12x + 6 - 1x - 1 = 2 44. 13 - x + 1x + 2 = 3
B Apply 45. Think About a Plan A hexagonal tray of vegetables has an area of 450 cm2.
What is the length of each side of the hexagon?
• What is the area of the triangle at the bottom in terms of the side length?
• How can you use the diagram at the right to find the formula for the area
of the hexagon? (Hint: Six triangles make one hexagon.)
46. Traffic Signs A stop sign is a regular octagon, formed by cutting triangles
off the corners of a square. If a stop sign measures 36 in. from top to bottom,
what is the length of each side?
Lesson 6-5 Solving Square Root and Other Radical Equations 395
47. Mental Math What is the solution? 1x + 11 = 4
48. You can find the area A of a square whose side is s units with the formula A = s2. What
is the best estimate for the side of a square with an area of 32 m2?
4.2 m 8.0 m
5.7 m 16 m
Solve. Check for extraneous solutions. 1
49. 3 12x - 3 = 9 50. 2(2x)3 + 1 = 5
51. 12x - 1 - 3 = 0 1
53. 2x2 + 3 = x + 1
52. (2x + 3)2 - 7 = 0
4
3
55. 2(x - 1)3 + 4 = 36
54. (2x + 3)4 - 3 = 5
57. 1x = 1x - 8 + 2
11
56. x2 - (x - 5)2 = 2
58. (x - 3)23 = x - 7
59. Error Analysis A student said that 4 and 1 are the solutions of the √x + 2 = x
problem shown. Describe and correct the student’s error.
√x = x - 2
STEM 60. Physics The velocity v of an object dropped from a tall building (√x)2 = (x - 2)2
is given by the formula v = 164d, where d is the distance the object
has dropped. Solve the formula for d. x = x2 - 4x + 4
0 = x2 - 5x + 4
61. Open-Ended Write an equation that has two radical expressions and no
real roots. 0 = (x - 4)(x - 1)
62. Reasoning You have solved equations containing square roots by squaring
each side. You were using the property that if a = b then a2 = b2. Show that the
following statements are not true for all real numbers.
a. If a2 = b2 then a = b.
b. If a … b then a2 … b2.
63. A teacher asked students why it is necessary to check for extraneous roots when
squaring both sides of the equation. Which of the following answers is the best? Is
this answer complete? Explain.
Because the squared equation can have negative roots.
B ecause squaring is multiplication, and any multiplication is a potential
source of extraneous roots.
Because when you square both sides of the equation a = b, you add to
the solution set the roots of the equation a = -b.
Because any operation with an equation may result in extraneous roots.
Solve. Check for extraneous solutions. 65. 2x + 12x = 12x
64. 1x + 1 + 12x = 15x + 3 67. 2 1x + 25 = 1x + 5
66. 2x + 12x = 2
396 Chapter 6 Radical Functions and Rational Exponents
C Challenge 68. Reasoning D evise a plan to find the value of x.
x = 52 + 22 + 12 + g
For each set of values, determine which is greater without using a calculator.
69. 16 or 12 + 1 70. 13 + 111 or 5
71. 110 or 12 + 13 72. 119 + 13 or 15 + 113
Standardized Test Prep
S AT/ACT 3
73. What is the solution of (x + 2)4 = 27?
x = 27 x = 79 x = 81 x = 83
74. A problem on a test asked students to solve a fifth-degree polynomial equation with
2i + 6,
rational coefficients. Adam found the following roots: -11.5, 12, 2 -12
and 3 - i. His teacher wrote that four of these roots are correct,
and one is incorrect.
Short Which root is incorrect?
R esponse
-11.5 12 2i + 6 3 - i
2
75. Which expression represents the solution of the equation x = a c b solved for a?
y +
bc - xy a yc yc + b yc - xb
+ x x
b
76. To rationalize the denominator of 44 , by what number would you multiply the
525
numerator and denominator of the fraction?
PERFORMANCE TASK
Apply What You’ve Learned MATHEMATICAL
PRACTICES
MP 6
Look back at the information on page 359 about the America’s Cup rule and
the dimensions for the new yacht you are designing. In the Apply What You’ve
Learned in Lesson 6-4, you wrote the expression on the left side of the America’s
Cup rule in radical form. Now, you will use this form of the expression to write a
function f (D) that gives the value of the expression for the new yacht in terms of
the displacement D.
a. To write the function, what variables will you replace with specific values? What are
those values?
b. Write and simplify the function.
c. Describe how you could use the function to write and solve an equation to find
the displacement that makes the value of the left side of the America’s Cup rule
exactly 24.
Lesson 6-5 Solving Square Root and Other Radical Equations 397
6-6 Function Operations CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
MF-BAFF.SA..911b2 .CF-oBmFb.1in.1ebs taCnodmarbdinfeunsctatinodnatrydpfeusnucstionng types
aursitnhgmaertiitchmopeetricatoiopnesra. tions.
FM-BAFF.SA..911c2 C.Fo-mBFp.o1s.e1cfu nCcotimonpso.se functions.
MP 1, MP 2, MP 3, MP 4
Objectives To add, subtract, multiply, and divide functions
To find the composite of two functions
Solve a simpler You want to buy a sofa that has already been
problem first. Use a marked down by $100. The furniture store
value for the cost of may add the 5% sales tax before applying
the sofa. the additional discount, or it may add the
sales tax after applying the additional
discount. Which way is better for you,
the customer? How much better?
MATHEMATICAL The final cost of the sofa in the Solve It involves two functions: one that gives an
additional discount and one that multiplies to find the sales tax.
PRACTICES
Essential Understanding You can add, subtract, multiply, and divide functions
Lesson based on how you perform these operations for real numbers. One difference, however,
Vocabulary is that you must consider the domain of each function.
• composite
function
Key Concepts Function Operations
Addition ( f + g)(x) = f (x) + g (x)
Subtraction # #( f - g)(x) = f(x) - g(x)
Multiplication
( f g)(x) = f (x) g (x)
Division f b (x) = gf ((xx)), g (x) ≠ 0
ag
The domains of the sum, difference, product, and quotient functions consist of the
x-values that are in the domains of both f and g. Also, the domain of the quotient
function does not contain any x-value for which g (x) = 0.
398 Chapter 6 Radical Functions and Rational Exponents
What determines the Problem 1 Adding and Subtracting Functions
domain of g?
Because there is a square Let f(x) = 4x + 7 and g(x) = 1x + x. What are f + g and f − g? What are their
root of x, x must be Ú 0. domains?
( f + g)(x) = f (x) + g (x) = (4x + 7) + ( 1x + x) = 5x + 1x + 7
( f - g)(x) = f (x) - g (x) = (4x + 7) - ( 1x + x) = 3x - 1x + 7
The domain of f is the set of all real numbers. The domain of g is all x Ú 0. The domain
of both f + g and f - g is the set of numbers common to the domains of both f and g,
which is all x Ú 0.
Got It? 1. Let f (x) = 2x2 + 8 and g (x) = x - 3. What are f + g and f - g? What are
their domains?
Problem 2 Multiplying and Dividing Functions
#Let f(x) = x2 − 9 and g(x) = x + 3. What are f
# #( f g)(x) = f(x) g(x) = (x2 - 9)(x + 3) g and f and their domains?
g
= x3 + 3x2 - 9x - 27
Is the domain of f f (x) - 3)(x +
g g (x) x+3
the domain of x − 3? #f b (x) = = x2 - 9 = (x 3) = x - 3, x ≠ -3
x+3
No; the fraction can only ag
The domain of both f and g is the set of real numbers, so the domain of f g is also the
be simplified and the
function is only defined set of real numbers.
when g(x) ≠ - 3.
The domain of f is the set of all real numbers except x ≠ -3, because g ( - 3) = 0.
The g requires that you consider the zero denominator in the original
f
definition of g
expression for f (x) despite the fact that the simplified form has the domain all real
g (x)
numbers.
#Got It? 2. Let f (x) = 3x2 - 11x - 4 and g(x) = 3x + 1. What are f g and f and
g
their domains?
The diagram shows what happens when Domain of f Range of f
you apply one function g (x) after another
function f (x). Input Output
of f of f
The output from the first function becomes x f(x)
the input for the second function. When you
combine two functions as in the diagram, Input Output
you form a composite function. of g of g
f(x) g(f(x))
Domain of g Range of g
Lesson 6-6 Function Operations 399
Key Concept Composition of Functions
The composition of function g with function f is written as g ∘ f and is defined as
(g ∘ f )(x) = g ( f (x)). The domain of g ∘ f consists of the x-values in the domain of f
for which f (x) is in the domain of g.
(g ∘ f )(x) = g ( f (x)) 1. Evaluate f(x) first.
1 2. Then use f(x) as the input for g.
2
Function composition is not commutative since f (g(x)) does not always equal g ( f (x)).
Problem 3 Composing Functions
Let f (x) = x − 5 and g (x) = x2. What is (g ∘ f )(−3)?
Method 1 Method 2 64
Which function is (g ∘ f )(x) = g ( f (x)) (g ∘ f )( -3) = g ( f ( -3)) 00000 0
11111 1
substituted into the = g (x - 5) = (x - 5)2 = g ( -3 - 5) 22222 2
33333 3
other? (g ∘ f )( -3) = ( -3 - 5)2 = g ( -8) 44444 4
Use f (x) as the 55555 5
input for g. 66666 6
= ( - 8)2 = ( - 8)2 77777 7
88888 8
= 64 = 64 99999 9
Got It? 3. What is ( f ∘ g)( -3) for the functions f and g defined in Problem 3?
Problem 4 Using Composite Functions
You have a coupon good for $5 off the price of any large pizza. You also get a 10%
discount on any pizza if you show your student ID. How much more would you pay
for a large pizza if the cashier applies the coupon first?
The coupon value and the The difference between • Compose two functions in two ways.
discount rate the results of applying the • T hen find the difference in their
discount or coupon first
results.
Step 1 Find functions C and D that model the cost of a large pizza.
Let x = the price of a large pizza.
Cost using the coupon: C (x) = x - 5
Cost using the 10% discount: D(x) = x - 0.1x = 0.9x
Step 2 Compose the functions to apply the discount and then the coupon.
(C ∘ D)(x) = C(D(x)) Apply the discount, D(x), first.
= C(0.9x) = 0.9x - 5
400 Chapter 6 Radical Functions and Rational Exponents
Step 3 Compose the functions to apply the coupon and then the discount.
(D ∘ C)(x) = D(C(x)) Apply the coupon, C (x), first.
= D(x - 5) = 0.9(x - 5) = 0.9x - 4.5
Step 4 Subtract the functions to find how much more you would pay if the cashier
applies the coupon first.
(D ∘ C)(x) - (C ∘ D)(x) = (0.9x - 4.5) - (0.9x - 5)
= -4.5 + 5
= 0.5
You pay $.50 more if the cashier applies the coupon first.
Got It? 4. A store is offering a 15% discount on all items. Also, employees get a 20%
employee discount. Write composite functions
a. to model taking the 15% discount and then the 20% discount.
b. to model taking the 20% discount and then the 15% discount.
c. Reasoning If you were an employee, which discount would you take
first? Why?
Lesson Check
Do you know HOW? Do you UNDERSTAND? MATHEMATICAL
PRACTICES
Let f (x) = 3x − 2 and g (x) = x2 + 1. Perform each 7. Error Analysis Your friend used some simple
#function operation. 2. ( f - g)(x) functions and found that ( f ∘ g)(x) = (g ∘ f )(x), and
concluded that function composition is commutative.
1. ( f g)(x)
Give an example to show that your friend is mistaken.
3. ( f ∘ g)(x)
4. f (x) + g (x) 8. Open-Ended F ind two functions f and g such that
f (g (x)) = x for all real numbers x.
5. g (x) - f (x) 6. f (x) - g (x)
Practice and Problem-Solving Exercises MATHEMATICAL
PRACTICES
A Practice Let f(x) = 7x + 5 and g(x) = x2. Perform each function operation and See Problems 1 and 2.
then find the domain of the result.
9. ( f + g)(x) 10. ( f - g)(x) 11. (g - f )(x)
13. gf (x)
# 12. ( f g)(x) 14. g (x)
f
Let f (x) = 2 − x and g (x) = 1x. Perform each function operation and then find
the domain of the result.
15. ( f + g)(x) 16. ( f - g)(x) 17. (g - f )(x)
19. gf (x)
# 18. ( f g)(x) 20. g (x)
f
Lesson 6-6 Function Operations 401
Let f (x) = 2x2 + x − 3 and g (x) = x − 1. Perform each function operation
and then find the domain.
21. ( f + g)(x) 22. ( f - g)(x) 23. (g - f )(x)
# 24. (f g)(x) 25. f (x) 26. g (x)
g f
Let g (x) = 2x and h (x) = x2 + 4. Find each value or expression. See Problem 3.
27. (h ∘ g)(1) 28. (h ∘ g)( -5) 29. (h ∘ g)( -2)
30. (g ∘ h)( -2) 31. (g ∘ h)(0) 32. (g ∘ h)(a)
33. (g ∘ g)(a) 34. (h ∘ h)(a) 35. (h ∘ g)(a)
Let f (x) = x2 and g (x) = x − 3. Find each value or expression.
36. (g ∘ f )( -2) 37. ( f ∘ g)( -2) 38. (g ∘ f )(0)
41. ( f ∘ g)(3.5)
39. ( f ∘ g)(0) 40. (g ∘ f )(3.5) 44. ( f ∘ g)( -a)
42. ( f ∘ g)(a) 43. (g ∘ f )( -a)
45. Sales A computer store offers a 5% discount off the list price x for any computer See Problem 4.
bought with cash, rather than put on credit. At the same time, the manufacturer
offers a $200 rebate for each purchase of a computer.
a. Write a function f (x) to represent the price after the cash discount.
b. Write a function g (x) to represent the price after the $200 rebate.
c. Suppose the list price of a computer is $1500. Use a composite function to find
the price of the computer if the discount is applied before the rebate.
d. Suppose the list price of a computer is $1500. Use a composite function to find
the price of the computer if the rebate is applied before the discount.
46. Economics S uppose the function f (x) = 0.15x represents the number of U.S.
dollars equivalent to x Chinese yuan and the function g (y) = 14.07y represents the
number of Mexican pesos equivalent to y U.S. dollars.
a. Write a composite function that represents the number of Mexican pesos
equivalent to x Chinese yuan.
b. Find the value in Mexican pesos of an item that costs 15 Chinese yuan.
Let f (x) = 2x + 5 and g (x) = x2 − 3x + 2. Perform each function operation
and then find the domain.
47. f (x) + g (x) 48. 3f (x) - 2 49. g (x) - f (x)
50. -2g (x) + f (x) 51. f (x) - g (x) + 10 #52. 4f(x) + 2g(x)
53. -f (x) + 4g (x) 54. f (x) - 2g (x) 55. f (x) g (x)
# 56. -3f(x) g(x) 57. f (x) 58. 5f (x)
g (x) g (x)
402 Chapter 6 Radical Functions and Rational Exponents
B Apply 59. Think About a Plan A craftsman makes and sells violins. The function
I (x) = 5995x represents the income in dollars from selling x violins. The function
P( y) = y - 100,000 represents his profit in dollars if he makes an income of
y dollars. What is the profit from selling 30 violins?
• How can you write a composite function to represent the craftsman’s profit?
• How can you use the composite function to find the profit earned when he sells
30 violins?
60. Suppose your teacher offers to give the whole class a bonus if everyone passes
the next math test. The teacher says she will give everyone a 10-point bonus and
increase everyone’s grade by 9% of their score.
a. You earned a 75 on the test. Would you rather have the 10-point bonus first and
then the 9% increase, or the 9% increase first and then the 10-point bonus?
b. Reasoning Is this the best plan for all students? Explain.
61. Sales A salesperson earns a 3% bonus on weekly sales over $5000. Consider the
following functions.
g (x) = 0.03x h (x) = x - 5000
a. Explain what each function above represents.
b. Which composition, (h ∘ g)(x) or (g ∘ h)(x), represents the weekly bonus?
Explain.
62. If ( f ∘ g)(x) = x2 - 6x + 8 and g (x) = x - 3, what is f (x)?
Let g (x) = 3x + 2 and f (x) = x − 2. Find each value.
3
63. f (g (1)) 64. g ( f ( -4)) 65. f (g (0)) 66. g ( f (2))
70. ( f ∘ f )(0)
67. g (g (0)) 68. (g ∘ g)(1) 69. ( f ∘ g)( -2)
71. Geometry Y ou toss a pebble into a pool of water and watch the circular ripples
radiate outward. You find that the function r (x) = 12.5x describes the radius r, in
inches, of a circle x seconds after it was formed. The function A (x) = px2 describes
the area A of a circle with radius x.
a. Find (A ∘ r)(x) when x = 2. Interpret your answer.
b. Find the area of a circle 4 seconds after it was formed.
For each pair of functions, find f(g (x)) and g (f (x)).
72. f (x) = 3x, g (x) = x2 73. f (x) = x + 3, g (x) = x - 5
74. f (x) = 3x2 + 2, g (x) = 2x
76. f (x) = -x - 7, g (x) = 4x 75. f (x) = x - 3, g (x) = 2x - 3
2
x + 5,
77. f (x) = 2 g (x) = x2
78. Open-Ended W rite a function rule that approximates each value.
a. The amount you save is a percent of what you earn. (You choose the percent.)
b. The amount you earn depends on how many hours you work. (You choose the
hourly wage.)
c. Write and simplify a composite function that expresses your savings as a
function of the number of hours you work. Interpret your results.
Lesson 6-6 Function Operations 403
C Challenge Let f (x) = x4 + 2x3 − 5x2 − 10x and g (x) = x3 − 3x2 − 5x + 15. Perform
#each function operation and simplify, and then find the domain.
79. f (x) g (x) 80. f (x) 81. g (x)
g (x) f (x)
Find each composition of functions. Simplify your answer.
82. Let f (x) = 1x. Find f ( f ( f (x))).
83. Let f (x) = 2x - 3. Find f (1 + h) - f (1), h ≠ 0.
h
84. Let f (x) = 4x - 1. Find f (a + h) - f (a), h ≠ 0.
h
85. Let f (x) = 4x2 - 1. Find f (a + h) - f (a), h ≠ 0.
h
Standardized Test Prep
S AT/ACT 86. Let f (x) = x + 5 and g (x) = x2 - 25. What is the domain of gf (x)?
All real numbers All real numbers except -5
All real numbers except 5 All real numbers except -5 and 5
87. Let g (x) = x - 3 and h (x) = x2 + 6. What is (h ∘ g)(1)?
-14 4 10 15
88. Which number is a solution of 0 3 - 2x 0 6 5? 4
-6 -1 2
S hort 89. What is the coefficient of the x3y4 term in the expansion of (3x - y)7? Show your
Response work.
Mixed Review
Solve. Check for extraneous solutions. See Lesson 6-5.
90. 2x2 + 3 = x + 1 91. x + 8 = 1x2 + 1 92. 2x2 + 9 = x + 1
16 2 2 11
93. 1x2 - 9212 - x = - 3 94. 2x2 + 12 - 2 = x 95. (3x)2 = (x + 6)2
Expand each binomial. 98. (2x - y)4 See Lesson 5-7.
96. (x + 4)8 97. (x + y)6 99. (2x - 3y)7
100. (9 - 2x)5 101. (4x - y)5 102. 1x2 + x24
103. 1x2 + 2y326
Get Ready! To prepare for Lesson 6-7, do Exercises 104–106.
Graph and solve each system. See Lesson 3-1.
104. eyy == xx -+ 66
105. e y = 0.5x + 21 106. y = x+4 4
y = 2x - ey = 5
5x -
404 Chapter 6 Radical Functions and Rational Exponents
6-7 Inverse Relations and MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Functions
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MMPAF1S,.M91P2.2F,-MBFP.23.4d
MP 1, MP 2, MP 3
Objective To find the inverse of a relation or function
What is wrong with the headline? Mayor’s Salary Restored
Why? What headline would you
have written? At last night’s meeting, the town
council approved a 20% increase in the
mayor’s salary. This follows last year’s
20% decrease.The Mayor’s comment was
Lesson If a relation pairs element a of its domain to element b of its range, the inverse relation
pairs b with a. So, if (a, b) is an ordered pair of a relation, then (b, a) is an ordered pair
Vocabulary of its inverse. If both a relation and its inverse happen to be functions, they are inverse
• inverse relation functions.
• inverse function
• one-to-one Essential Understanding The inverse of a function may or may not be a function.
function
This diagram shows a relation r (a function) and its Relation r Inverse of r
inverse (not a function). The range of the relation is the
domain of the inverse. The domain of the relation is the Domain Range Domain Range
range of the inverse.
1.2 1 1 1.2
1.4 1.4
1.6 1.6
1.9 2 2 1.9
Pro blem 1 Finding the Inverse of a Relation
(0, −1) is in s. How A What is the inverse ofIrnevlaetrisoenosf? Inverse of
do you find the Relation s Relation s Relation s Relation s
corresponding pair in
the inverse of s? xy xy xy x y
0 Ϫ1 0
Switch the coordinates. 20 SwitcϪh t1he x- 0and y-values to g0et theϪin1verse. Ϫ1 2
( - 1, 0) is in the inverse 32 02 20 0 3
of s. 43 2 4
3
23 32
34 43
Lesson 6-7 Inverse Relations and Functions 405
B What are the graphs of s and its inverse?
Relation s Reversing Inverse of s
4y the Ordered Pairs 4y
4y
2 22
x (؊1, 0) x x
Ϫ4 Ϫ2 O 2 4 Ϫ4 Ϫ2 O 2 4 Ϫ4 Ϫ2 O 2 4
Ϫ2 Ϫ2 (0, ؊1) Ϫ2
Got It? 1. a. What are the graphs of t and its inverse? Relation t
b. Reasoning Is t a function? Is the inverse
x 0123
of t a function? Explain. y Ϫ5 Ϫ4 Ϫ3 Ϫ3
As shown in Problem 1, the graphs of a relation and its inverse are the reflections of
each other in the line y = x. If you describe a relation or function by an equation in
x and y, you can switch x and y to get an equation for the inverse.
Problem 2 Finding an Equation for the Inverse
Why do you solve What is the inverse of the relation described by y = x 2 − 1?
for y?
If you solve the equation y = x2 - 1
for y, you can use it to
easily generate ordered x = y 2 - 1 Switch x and y.
pairs that are part of the
inverse relation. x + 1 = y2 Add 1 to each side.
{ 1x + 1 = y Find the square root of each side to solve for y.
Got It? 2. What is the inverse of y = 2x + 8?
Problem 3 Graphing a Relation and Its Inverse
What does the
graph of y = x 2 − 1 What are the graphs of y = x2 − 1 and its inverse, y = t 1x + 1?
look like?
The graph of y = x2 - 1 The graph of y = x2 - 1 is a parabola that opens y
is a translation of y = x2 upward with vertex (0, -1). The graph of the inverse
d own one unit. is the reflection of the parabola in the line y = x. y ؍x2 ؊ 1
Got It? 3. What are the graphs of y = 2x + 8 and 2 y ؎؍x؉1
its inverse?
Ϫ2 2 4 6x
Ϫ2
406 Chapter 6 Radical Functions and Rational Exponents
The inverse of a function f is denoted by f -1. You read f -1 as “the inverse of f ”or as
“f inverse.” The notation f (x) is used for functions, but the relation f -1 may not even be
a function.
Problem 4 Finding an Inverse Function
Consider the function f (x) = 1x - 2.
Problem 1
A What are the domain and range of f ?
The radicand cannot be negative, so the numbers x Ú 2 make up the domain. The
principal square root is nonnegative, so the numbers y Ú 0 make up the range.
B What is f −1, the inverse of f ?
f (x) = 1x - 2
y = 1x - 2 Rewrite the equation using y.
x = 1y - 2 Switch x and y. Since x equals a principal square root, x Ú 0.
x2 = y - 2 Square both sides.
y = x2 + 2 Solve for y.
So, f -1(x) = x2 + 2, for x Ú 0.
C What are the domain and range of f −1?
H ow could a graph Part (b) shows that the domain of f -1 is the range of f—the numbers x Ú 0. Since
x2 Ú 0, x2 + 2 Ú 2. Therefore, the numbers y Ú 2 make up the range of f -1. Note
help you check your that the range of f -1 is the same as the domain of f.
D Is f −1 a function? Explain.
answer?
You could graph f - 1 and
s ee whether the graph
fpt ueasnstcs.teIifsointth.deoveesr,tfic-al1liisnea Ffo-r1e(xa)ch= xx2in+th2e, xdoÚm0a,iins a(xfuÚnc0t)ioonf .f -1, there is only one value of y in the range. So
Got It? 4. Let g(x) = 6 - 4x.
a. What are the domain and range of g?
b. What is the inverse of g?
c. What are the domain and range of g-1?
d. Is g-1 a function? Explain.
Functions that model real-world behavior are often expressed as formulas with
meaningful variables, like A = pr2 for the area of a circle. Strictly speaking, the inverse
formula would be r2 = pA, but this expresses a false relationship between A and r. It is
better to leave the variables in place and solve for r as a function of A.
A = pr2 Original formula.
r = 5pA Same formula, but inversely expressed.
Lesson 6-7 Inverse Relations and Functions 407
Problem 5 Finding the Inverse of a Formula
Why shouldn’t you The function d = 4.9t2 represents the distance d,
interchange the in meters, that an object falls in t seconds due to
variables? Earth’s gravity. Find the inverse of this function.
Interchanging the How long, in seconds, does it take for the cliff diver
variables leads to a false shown to reach the water below?
relationship between
distance and time. d = 4.9t2
t2 = d Solve for t. the variables.
4.9 Do not switch
t = 54d.9 Time must be nonnegative.
= 24 Substitute 24 for d. 24 meters
5 4.9
≈ 2.2 Use a calculator.
It will take about 2.2 seconds for the diver to reach
the water.
Got It? 5. The function d = v2 relates the distance d, in meters, that an object
19.6
has fallen to its velocity v, in meters per second. Find the inverse of this
function. What is the velocity of the cliff diver in meters per second as he
enters the water?
You know that for any function f, each x-value in the domain corresponds to exactly
one y-value in the range. For a one-to-one function, it is also true that each y-value
in the range corresponds to exactly one x-value in the domain. A one-to-one function
f has an inverse f -1 that is also a function. If f maps a to b, then f -1 must map b to a.
Domain of f Range of f
Range of f ؊1 Domain of f ؊1
f
ab
f Ϫ1
Key Concept Composition of Inverse Functions
If f and f -1 are inverse functions, then
( f -1 ∘ f )(x) = x and ( f ∘ f -1)(x) = x for x in the domains of f and f -1, respectively.
This says that the composition of a function and its inverse is essentially the identity
function, id (x) = x, or y = x.
408 Chapter 6 Radical Functions and Rational Exponents
Problem 6 Composing Inverse Functions
For f (x) = x 1 1, what is each of the following?
Is this a function? −
Yes. For each value of x,
there is only one value A f −1(x)
for y.
f (x) = x 1 1
-
1
y = x - 1 Rewrite the equation using y.
x = y 1 1 Switch x and y.
-
x( y - 1) = 1 Solve for y.
y - 1 = 1
x
y = 1 + 1
x
So f -1(x) = 1 + 1.
x
B ( f ° f −1)(1) C ( f −1 ° f )(1)
( f ∘ f -1)(1) = f ( f -1(1)) ( f -1 ∘ f )(1) = f -1( f (1))
( )= f 1 + 1 ( )= f-1 1 1
1 -1
= f (2) -1( )= f 1 undefined
0
= 2 1 1 = 1 1 is not in the domain of f. Therefore
- ( f -1 ∘ f )(1) does not exist.
Got It? 6. Lae. tgg-(1x()x)= x +4 2. What is eabc.h (ogf∘thge-1f)o(l0lo) wing? c. (g-1 ∘ g)(0)
Lesson Check
Do you know HOW? Do you UNDERSTAND? MATHEMATICAL
Find the inverse of each function. Is the inverse a PRACTICES
function?
5. Vocabulary Does every function have an inverse
which is a function? Does every relation have an
1. f (x) = 4x + 3 inverse which is a relation?
2. f (x) = x2 - 1
3. f (x) = (x + 1)2 6. Reasoning A function consists of the pairs (2, 3), (x, 4),
and (5, 6). What values, if any, may x not assume?
4. For h(x) = -x 1 2 , find: 7. Error Analysis A classmate says that
a. h-1(x) + ( f ∘ g)-1(x) = ( f -1 ∘ g-1)(x). Show that this is
incorrect by finding examples of f (x) and g (x) for
b. h-1(4)
which the equation does not hold.
c. Value of x for which the equality (h ∘ h-1)(x) = x
does not hold.
Lesson 6-7 Inverse Relations and Functions 409
Practice and Problem-Solving Exercises MATHEMATICAL
PRACTICES
A Practice Find the inverse of each relation. Graph the given relation and its inverse. See Problem 1.
8. x y 9. x y 10. x y 11. x y
10 1 0 0 0 Ϫ3 2
21 2 1 1 1 Ϫ2 2
30 3 2 2 4 Ϫ1 2
42 4 3 3 9 0 2
Find the inverse of each function. Is the inverse a function? See Problem 2.
12. y = 3x + 1 13. y = 2x - 1 14. y = 4 - 3x
17. y = 3x2 - 5
15. y = 5 - 2x2 16. y = x2 + 4 20. y = (1 - 2x)2 + 5
18. y = (x - 8)2 19. y = (3x - 4)2
Graph each relation and its inverse. See Problem 3.
21. y = 2x - 3 22. y = 3 - 7x 23. y = -x
24. y = 3x2 25. y = - x2 26. y = 4x2 - 2
29. y = (3 - 2x)2 - 1
27. y = (x - 1)2 28. y = (2 - x)2
For each function, find the inverse and the domain and range of the function See Problem 4.
and its inverse. Determine whether the inverse is a function.
30. f (x) = 3x + 4 31. f (x) = 1x - 5
32. f (x) = 1x + 7 33. f (x) = 1-2x + 3
34. f (x) = 2x2 + 2 35. f (x) = -x2 + 1
36. Temperature The formula for converting from Celsius to Fahrenheit See Problem 5.
temperatures is F = 95C + 32.
a. Find the inverse of the formula. Is the inverse a function?
b. Use the inverse to find the Celsius temperature that corresponds to 25°F.
37. Geometry T he formula for the volume of a sphere is V = 43pr3.
a. Find the inverse of the formula. Is the inverse a function?
b. Use the inverse to find the radius of a sphere that has a volume of 35,000 ft 3.
For Exercises 38–41, f(x) = 10x − 10. Find each value. See Problem 6.
38. ( f -1 ∘ f )(10) 39. ( f ∘ f -1)( - 10)
40. ( f -1 ∘ f )(0.2) 41. ( f ∘ f -1)(d)
410 Chapter 6 Radical Functions and Rational Exponents
B Apply Find the inverse of each function. Is the inverse a function?
42. f (x) = x3 43. f (x) = x4 44. f (x) = 2x2 + 1
45. f (x) = 1.5x2 - 4 5
3x2
46. f (x) = 4 47. f (x) = 12x - 1 + 3
48. Think About a Plan The velocity of the water that flows from an opening at the
base of a tank depends on the height of water above the opening. The function
v (x) = 12gx models the velocity v in feet per second where g, the acceleration due
to gravity, is about 32 ft>s2 and x is the height in feet of the water. What is the depth
of water when the flow is 40 ft/s, and when the flow is 20 ft/s?
• How can you use inverse functions to help you find the answer?
• What restrictions are on the domain of v(x)? of v-1(x)?
49. Let f (x) = 3x2 - 4 and g (x) = x - 2. Calculate ( f ∘ g-1)(x) for x = - 3.
50. Writing E xplain how you can find the range of the inverse of f (x) = 1x - 1
without finding the inverse itself.
For each function, find the inverse and the domain and range of the function
and its inverse. Determine whether the inverse is a function.
51. f (x) = - 1x 52. f (x) = 1x + 3 53. f (x) = 1-x + 3
54. f(x) = 1x + 2
57. f (x) = (x - 4)2 55. f(x) = x2 56. f (x) = 1
2 x2
60. f (x) = 4 - 21x 1
58. f (x) = (7 - x)2 59. f (x) = (x + 1)2
61. f (x) = 3 62. f (x) = 1
1x 1 - 2x
63. a. O pen-Ended C opy the mapping diagram at the right. Complete it Relation r
by writing members of the domain and range and connecting them Domain Range
with arrows so that r is a function and r-1 is not a function.
b. Repeat part (a) so that r is not a function and r-1 is a function.
64. Reasoning R elation r has one element in its domain and two elements in
its range. Is r a function? Is the inverse of r a function? Explain.
65. Geometry W rite a function that gives the length of the hypotenuse of an isosceles
right triangle with side length s. Evaluate the inverse of the function to find the side
length of an isosceles right triangle with a hypotenuse of 6 in.
66. For the function f (x) = 13 2x, find f -1(x). Then determine the value of x when f (x) = 16.
67. Reasoning To determine if the inverse of function f is also a function, you can use
a horizontal line test. It says that if no horizontal line intersects the graph of the
function f in more than one point, then the inverse of f is a function.
a. Explain why the horizontal line test works.
b. The graph of a polynomial function passes through the points ( -1, 1), (0, 4) and (2, 3).
Can its inverse be a function?
Lesson 6-7 Inverse Relations and Functions 411
C Challenge Find the inverse of each function. Is the inverse a function? 13 x
3
68. f (x) = 15 x 3 69. f (x) = 13 x - 5 70. f (x) =
71. f (x) = (x - 2)3 72. f (x) = 14 x
73. f (x) = 1.2x4
74. Function f (x) is defined the following way:
• if x is an integer, then f (x) = x + 1;
• for all other x, f (x) = x + 2.
Is the inverse of f (x) a function? Explain.
Standardized Test Prep
S AT/ACT 75. Which pair of words makes this sentence FALSE?
The product of two ____(I)____ numbers is always a (n) ____(II)____ number.
(I) complex; (II) complex (I) rational; (II) real
(I) real; (II) complex (I) imaginary; (II) imaginary
76. If f (x) = x + 1 and g (x) = x2 - 3x - 4, what is ( f ∘ g)(x)?
x2 - 3x - 3 x2 - x - 6 x2 - x x2 - x - 3
2 32
a3b4 ?
( )
77. What is the simplified form of
49 43 17
a9b 16 a3b2 ab (ab) 6
E xtended 78. Let f (x) = (x + 1)2 - 2. Find the x- and y-intercepts of f (x) and the inverse of
Response f (x). Is the inverse a function?
Mixed Review
Let f (x) = 4x, g(x) = 12x + 7, and h (x) = −2x + 4 . Perform each function See Lesson 6-6.
operation.
79. (g ∘ f )(x) 80. (h ∘ g)(x) 81. h(x) + g(x)
83. ( f ∘ g)(x) + h(x) 84. ( f ∘ g)(x)
# 82. f (x) g (x)
Find each real root. See Lesson 6-1.
85. -14 16
89. 15 - 243 86. 14 - 16 87. 15 243 88. -15 243
90. 13 0.064 91. 14 810,000
92. 41
5160,000
Get Ready! To prepare for Lesson 6-8, do Exercises 93–95.
Graph each function. See Lesson 4-1.
93. y = - x2 - 1 94. y = -(x + 1)2 + 1 95. y = 3x2 + 3
412 Chapter 6 Radical Functions and Rational Exponents
Concept Byte Graphing Inverses CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
For Use With Lesson 6-7 Extends FM-BAFF.BS.49a1 2S.Fo-lvBeF.a2n.4eaq uSaotilovne oafntheequfoartmionf(oxf) =
cthfeorfoarmsimf(pxl)e=fucnfcotiroansfimthpalet hfuanscatnioinnvfetrhsaetahnads wanrite an
Technology einxvperressesiaonndfowrrtitheeainveexrpsere. ssion for the inverse.
MP 5
You can graph inverses of functions on a graphing calculator by using the DrawInv MATHEMATICAL
feature or by using parametric equations. It takes more keystrokes to set up parametric
equations, but once you do you can easily change from one function to another and PRACTICES
quickly see the graphs of the new function and its inverse.
Graph y = 0.3x2 + 1 and its inverse.
Method 1 Use the DrawInv feature.
Step 1 Press y= and enter the equation. Press zoom 5 to see a graph of the
function with equal x- and y-intervals.
Step 2 Press 2nd draw 8. You will see DrawInv followed by a flashing cursor.
Select equation Y1 by pressing vars 1 1. Press enter to see the graph of
the function and its inverse.
Method 2 Use parametric equations.
Step 1 Set to parametric mode. Press mode , select Par, and press 2nd quit . Plot1 Plot2 Plot3
\X1T = T Plot3
Step 2 Enter the given equation in parametric form. Press y= and enter the Y1T = .3T2 + 1
equations X1T = T and Y1T = .3T2 + 1. \\PYYXX\\\YYXXXl1212oTTT23T423tTTTTT1=========YX.T3YX11TTT112TTP+lo1t2
\X3T =
Step 3 Now use X2T = Y1T and Y2T = X1T to interchange the x- and y-values Y3T =
of the first parametric equation. Press y= and move the cursor to follow \X4T =
X2T =. Select Y1T by pressing vars 2 2. Enter the equation Y2T = X1T in
a similar fashion.
Step 4 Press zoom 5. Adjust the Window so that Tmin and Tmax approximately
agree with Xmin and Xmax. Press graph to see the graph of the function
and its inverse.
Exercises
Graph each function and its inverse with a graphing calculator. Then sketch the
graphs.
1. y = x2 - 5 2. y = (x - 3)2 3. y = 0.01x4 4. y = 0.5x3 - 3
5. Writing Change the parametric equation X2T = Y1T in Method 2, Step 3 to
X2T = - Y1T. Describe the graph that results.
6. Explain how once you set up parametric equations, you can change from one
function to another and quickly see the graphs of the new function and its inverse.
Concept Byte Graphing Inverses 413
6-8 Graphing Radical MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Functions
FM-IAF.FCS..79b1 2G.Fra-IpFh.3s.q7uba reGroaopthasnqudacreubroeorot oatnfdunccutiboensro.o. t.
Ffu-nIFc.tCio.8n sW. .ri.te a function defined by an expression in
MdifAfeFreSn.9t 1b2ut.Fe-qIFu.i3va.8le nWt froitremas f.u.n.ction defined by an
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MP 1, MP 2, MP 3, MP 4, MP 5
Objective To graph square root and other radical functions
Make a sketch so A red plastic strip binds the three
you can try out
ideas. identical cylinders. The cross-sectional
area enclosed by the strip is 115 cm2.
What is the radius of each cylinder?
What is the length of the plastic
strip?
MATHEMATICAL
PRACTICES
The formula A = pr2 shows that area is a quadratic function of the radius of a
= 1
Lesson circle. The formula r 1p 1A shows that the radius of a circle is a square root
function of the area.
Vocabulary
• radical function Essential Understanding A square root function is the inverse of a quadratic
• square root function that has a restricted domain.
function
A horizontal line can intersect the graph of f (x) = x2 in two points—where
f ( -2) = f (2), for example. Thus, a vertical line can intersect the graph of f -1 in two
points. f -1 is not a function because it fails the vertical line test.
y 2y
f(x) ؍x2 2 x O2 f؊1(x) ؎ ؍x
Ϫ4 Ϫ2 O 2 Ϫ2 x
However, you can restrict the domain of f so that the inverse of the restricted function
is a function.
y y
2
2 x
f(x) ؍x2, x Ն 0 2 f؊1(x) ؍x
O 24 x
Ϫ2 O
414 Chapter 6 Radical Functions and Rational Exponents
Inverses of the power functions y = xn (with domains restricted as needed) form
parent functions y = 1n x for families of radical functions. In particular, f (x) = 1x
is the parent for the family of square root functions. Members of this family have the
general form f (x) = a 1x - h + k.
Key Concepts Families of Radical Functions
Parent function: Square Root Radical
y = 1x y = 1n x
Reflection in x-axis: y = - 1x y = - 1n x
Stretch (a 7 1), shrink
y = a1x y = a 1n x
(0 6 a 6 1) by the factor a:
Translation: H orizontal by h y = 1x - h + k y = 1n x - h + k
Vertical by k
Problem 1 Translating a Square Root Function Vertically
How is y = 1x + k What are the graphs of y = 1x − 2 and y = 1x + 1?
related to the parent
function y = 1x? The graph of y = 1x - 2 is the graph of y = 1x y y ؍x؉1
It is related to the parent shifted down 2 units. 2 y ؍x
function in the same way The graph of y = 1x + 1 is the graph of y = 1x O 46 8 x
that y = f(x) + k is shifted up 1 unit. Ϫ2 y ؍x ؊ 2
related to y = f(x). It
is a vertical translation The domains of both functions are the set of
nonnegative numbers, but their ranges differ.
of k units.
Got It? 1. What are the graphs of y = 1x + 2 and y = 1x - 3?
Problem 2 Translating a Square Root Function Horizontally
What are the graphs of y = 1x + 4 and y = 1x − 1?
The graph of y = 1x + 4 is the graph of y = 1x y
shifted left 4 units.
How is y = 1x − h 4 y ؍x؉4
related to the parent The graph of y = 1x - 1 is the graph of y = 1x
function y = 1x? shifted right 1 unit. 2
It is a horizontal
The ranges of both functions are the set of y ؍x y ؍x؊1
translation of h units. nonnegative numbers, but their domains differ. Ϫ4 Ϫ2 O 24 x
Got It? 2. What are the graphs of y = 1x - 3 and y = 1x + 2?
Lesson 6-8 Graphing Radical Functions 415
Recall from Lesson 2-7 that for any transformation,y = af (x - h) + k of the parent
function f (x), a indicates a vertical stretch or shrink.
Similarly, for the combined transformation y = a1x - h + k, a indicates a vertical
stretch ( 0 a 0 7 1) or shrink ( 0 a 0 6 1). A negative value of a indicates a reflection in
the x-axis.
Problem 3 Graphing a Square Root Function
What would be good What is the graph of y = − 1 1x − 3 + 1? 4y
points to choose? 2 2 y ! ؍x
Points that have integer
x- and y-coordinates. Step 1 Choose several points from the parent Step 1
function y = 1x.
y ؍ ؊ 1 !x ؊ 3 ؉ 1
Step 2 Multiply the y-coordinates by 2
gara=p-h 12v.eTrhtiicsasllhyrbinyktshtehfeapctaorren12 tand
reflects the result in the x-axis. O 6x Step 3
Step 2
Ϫ2 y ؍ ؊ 1 !x
2
Step 3 The values of h and k give the
horizontal and vertical translations.
Translate the graph from Step 2
to the right 3 units and up 1 unit.
Got It? 3. What is the graph of y = 3 1x + 2 - 4?
Problem 4 Solving a Radical Equation by Graphing
Multiple Choice Y ou can model the population P of Corpus Christi, texas, between the
years 1970 and 2005 by the radical function P (x) = 75,000 13 x − 1950, where x is the
year. Using this model, in what year was the population of Corpus Christi 250,000?
1980 1983 1987 1990
For P = 250,000, solve the equation 250,000 = 75,000 13 x - 1950.
How can you rewrite Graph Y1 = 75000(X − 1950)^(1/3) and Y2 = 250000. Adjust the
window to find where the graphs intersect.
a radical function
using an exponent?
You can write a radical Use the Intersect feature to find the x-coordinate of the intersection.
function y = 1n x as
1 In the year 1987, the population of Corpus Christi was 250,000.
y = x n . The correct answer is C. Intersection
X=1987.04 Y=250000
Got It? 4. In what year was the population of Corpus Christi 275,000?
y y ! ؍3 x
x
Problem 4 uses a transformation of y = 13 x . The function 2
f (x) = 13 x is the inverse of g (x) = x 3. Unlike y = 1x, the domain
and range of f (x) = 13 x are all real numbers. Ϫ8 O4 8
416 Chapter 6 Radical Functions and Rational Exponents
The patterns for graphing square root functions apply to other radical functions.
Problem 5 Graphing a Cube Root Function Step 2
6y
What is the graph of y = 213 x + 1 − 4?
y ؍2!3 x
How is Step 1 Graph the parent function, y = 13 x.
y = a1n x − h + k
related to its parent Step 2 Multiply the y-coordinates by 2. 2
function? This stretches the graph vertically.
a stretches or shrinks
the parent function Step 3 Translate the graph from step 2, y ! ؍3 x x
and h and k translate 1 unit to the left and 4 units down.
it horizontally and Step 1 Ϫ6 Ϫ4 Ϫ2 O2 6
vertically.
y ؍2!3 x ؉ 1 ؊ 4
Got It? 5. What is the graph of Step 3 Ϫ6
y = 3 - 12 13 x - 2 ?
You can graph functions of the form y = 1n bx + c using transformations, if you can
simplify the radicand so that x has a coefficient of 1. This is also true for functions in the
form y = a1n bx + c + k.
Problem 6 Rewriting a Radical Function
How can you rewrite y = 19x + 18 so you can graph it using transformations?
Describe the graph.
The form y = a1x shows the y = 19x + 18
stretch or shrink. Factor to get y = 19(x + 2)
x - h in the radicand. y = 19(x − (−2))
Find the square root of 9. y = 31x − (−2)
Now, you have the form
y = a1x - h that you can The graph of y = 19x + 18 is the
graph using transformations. graph of y = 31x translated
2 units to the left.
Got It? 6. a. How can you rewrite y = 13 8x + 32 - 2 so you can graph it using
transformations? Describe the graph.
b. Reasoning Describe the graph of y = 0 9x + 18 0 by rewriting it in the
form y = a 0 x - h 0 . How is this similar to rewriting y = 19x + 18 in
Problem 6?
Lesson 6-8 Graphing Radical Functions 417
Lesson Check Do you UNDERSTAND? MATHEMATICAL
Do you know HOW? PRACTICES
Graph each function.
5. Writing E xplain the effect that a has on the graph
1. y = - 1x + 3 2. y = - 13 x + 5 of y = a1x. How does this compare to its effect on
other functions you have studied?
Rewrite each function so you can graph it using 6. Error Analysis Your friend states that the graph of
transformations of its parent function. Describe the graph.
the function g (x) = 1-x - 1 is a reflection of the
3. y = 14x - 4 4. y = 13 8x + 16 graph of the function f (x) = - 1x + 1 across the
x-axis. Describe your friend’s error.
Practice and Problem-Solving Exercises MATHEMATICAL
A Practice Graph each function. PRACTICES See Problems 1 and 2.
10. y = 1x + 5
7. y = 1x + 1 8. y = 1x - 2 9. y = 1x - 4 14. y = 1x - 4
11. y = 1x - 3 12. y = 1x + 1 13. y = 1x + 6
Graph each function. 16. y = - 1x - 1 See Problem 3.
15. y = 3 1x 17. y = -5 1x + 2
18. y = -0.5 1x + 3 19. y = 1 1x + 2 - 1 20. y = 31x + 1 + 4
2
Solve each square root equation by graphing. Round the answer to the nearest See Problem 4.
hundredth, if necessary. If there is no solution, explain why.
21. 1x - 3 = 12 22. 12x - 3 = 4 23. 12x + 5 = 12 - x
24. Landscaping A sprinkler can water between 1 and 130 square yards of a lawn. The
length L in inches of rotating pipe needed to water A square yards is given by the
function L = 117.75 1A.
a. Graph the equation on your calculator. Make a sketch of the graph.
b. How much area can be watered if the length of the pipe is 500, 800, or
1300 inches long?
Graph each function. See Problem 5.
25. y = 13 x + 5
28. y = - 13 x + 3 - 1 26. y = 13 x - 4 27. y = 13 x + 2 - 7
29. y = 2 13 x - 6 - 9
30. y = 1 13 x - 1 + 3
2
Rewrite each function to make it easy to graph using transformations of its See Problem 6.
parent function. Describe the graph.
31. y = 19x - 9 32. y = - 116x + 32 33. y = -2 14x + 16
34. y = 13 64x + 128 35. y = 125x + 125 - 3 36. y = 13 8x - 24 + 1
418 Chapter 6 Radical Functions and Rational Exponents
B Apply 37. Think About a Plan The time t in seconds for a pendulum to complete one full
cycle is given by the function t = 1.11 1l, where l is the length of the pendulum in
feet. How long is a pendulum that takes 4.5 seconds to complete one full cycle?
6 seconds to complete one full cycle? Round your answers to the nearest
hundredth.
• How can you use a graph to approximate the length of a pendulum?
• How can you check your answers algebraically?
Graph each function. Find the domain and range.
38. y = 4 13 x - 2 + 1 39. y = 1 1x - 1 + 3 40. y = 3 13 x - 6 + 2
2
41. Suppose that a function pairs elements from set A with elements from set B. 2y x
Recall that a function is called onto if every element in B is paired with at least
one element in A. O 2
Ϫ2
a. The graph shows a transformation of y = 1x. Write the function.
b. What are the domain and range of the function?
c. For the domain, is the function onto the set of nonnegative real numbers?
Explain.
42. Open-Ended Write a radical function such that for its domain, the function is onto
the set of real numbers such that y … 3.
Rewrite each function to make it easy to graph using transformations of its
parent function. Describe the graph.
43. y = 125x - 100 - 1 44. y = 136x + 108 + 4 45. y = - 13 8x - 2
46. y = 5x -4 1 - 2 47. y = 10 - 3 x + 3 48. y = x + 1 + 5
27 59
5
Graphing Calculator Solve the following radical equations.
49. 21x = 1(x + 1) 50. 1(x + 3) = 41(x) - 2 51. 13 x - 1 = 1x - 1
52. a. Solve 3 - 1(x - 3) = x algebraically.
b. Solve the equation from part (a) graphically.
c. What do you notice about your answer to part (a) compared to your answer to
part (b)?
STEM 53. Electronics T he size of a computer monitor is given as the length of the screen’s
5
diagonal d in inches. The equation d = 6 13A models the length of a diagonal of a
monitor screen with area A in square inches.
a. Graph the equation on your calculator.
b. Suppose you want to buy a new monitor with a screen that is twice the area of
your old screen. Your old screen has a diagonal of 15 inches. What will be the
diagonal of your new screen?
STEM 54. Physics You can model time t, in seconds, an object takes to reach the ground
falling from height H, in meters, by t (H) = 52gH. The value of g is 9.81 m>s2. If an
object takes 7 seconds to fall to the ground, what was its initial height?
Lesson 6-8 Graphing Radical Functions 419
C Challenge Rewrite each function to make it easy to graph using transformations of its
parent function. Describe the graph. Find the domain and range of each
function.
55. y = - 12(4x - 3) 56. y = 13x - 5 + 6 57. y = -3 - 112x + 18
58. a. Graph y = 1-x, y = 11 - x, and y = 12 - x.
b. Make a Conjecture How does the graph of y = 1h - x differ from the graph
of y = 1x - h?
59. For what positive integers n are the domain and range of y = 1n x the set of real
numbers? Assume that x is a real number.
PERFORMANCE TASK
Apply What You’ve Learned MATHEMATICAL
PRACTICES
MP 4, MP 5
In the Apply What You’ve Learned in Lesson 6-5, you wrote a function f (D)
for the new yacht, described on page 359. The function gives the value of the
expression on the left side of the America’s Cup rule for any displacement D.
Use a graphing calculator to graph this function. Select all of the following that
are true. Explain your reasoning.
A. For any viewing window, the graph of f (D) lies entirely above the x-axis.
B. In the context of this real-world situation, the relevant domain of the function is all
real numbers.
C. As the value of D increases, the value of f (D) decreases.
D. The graph of the function intersects the horizontal line y = 24 at exactly one point.
E. The graph shows that when D = 10, f (D) is less than 24.
F. The graph shows that a displacement of 30 cubic meters is one possible
displacement for the new yacht.
420 Chapter 6 Radical Functions and Rational Exponents
6 Pull It All Together
RMANCPERFOE TASKCompleting the Performance Task
To solve these Look back at your results from the Apply What You’ve Learned sections in Lessons 6-4,
problems, 6-5, and 6-8. Use the work you did to complete the following.
you will pull
together 1. Solve the problem in the Task Description on page 359 by verifying that the AC45
concepts and Wingsail Catamaran satisfies the America’s Cup rule, and by finding the possible
skills related displacements for your yacht. Show all your work and explain each step of your
to roots solution.
and radical
functions. 2. Reflect Choose one of the Mathematical Practices below and explain how you
applied it in your work on the Performance Task.
MP 2: Reason abstractly and quantitatively.
MP 4: Model with mathematics.
MP 5: Use appropriate tools strategically.
MP 6: Attend to precision.
On Your Own
A yacht designer is considering the dimensions shown below for a new yacht called the
SailSmart Catamaran.
SailSmart Catamaran
Length 17.35 meters
Sail Area 97.5 square meters
Displacement 2.1 cubic meters
a. Does the SailSmart Catamaran satisfy the rule on page 359? Explain.
b. The designer would like to change only the sail area so that the value of the
expression on the left side of the rule is at least 20 but no more than 24. Determine
a range of sail areas the designer could use to meet this goal.
Chapter 6 Pull It All Together 421
6 Chapter Review
Connecting and Answering the Essential Questions
1 Equivalence Radical Expressions and Rational Solving Square Root Equations
Exponents (Lessons 6-1, 6-2 and 6-4) (Lesson 6-5)
You can simplify the nth
23 - 8x5 23 x2 = 23 - 8x7 x - 2 = 1x
root of an expression that x2 - 4x + 4 = x
#= 23 (-2)3x6 x x2 - 5x + 4 = 0
contains an nth power as a (x - 4)(x - 1) = 0
x = 4 or x = 1
factor. 4 - 2 = 14 ✓
1 - 2 ≠ 11 ✗
2n xn = x n = x, n odd = - 2x2 13 x
n
0 x 0 , n even
# #( - 8x5)13 (x2)31 = ( - 8x7)13
2 solving Equations = (( - 2)3 x6 1
and Inequalities 1
When you square each side x)3
of an equation, the resulting x3
equation may have more = - 2x2
solutions than the original
equation. Inverse Relations and Functions Graphing Radical Functions
(Lesson 6-7) (Lesson 6-8)
3 Function
If f and f -1 are inverse The inverse of y = 1x + 2, x Ú 0, y Ú 2 y
functions and if one maps is x = 1y + 2, or 1y = x - 2, 6 y ( ؍x ؊ 2)2
a to b, then the other maps or y = (x - 2)2, y Ú 0, x Ú 2.
b to a, i.e., 4 y √ ؍x ؉ 2
(f ∘ f -1)(a) = (f -1 ∘ f )(a) 2
= a. x
O 2468
Chapter Vocabulary • one-to-one function (p. 408) • rationalize the
• principal root (p. 361) denominator (p. 369)
• composite function (p. 399) • radical equation (p. 390) • simplest form of a radical (p. 368)
• index (p. 362) • radical function (p. 415) • square root equation (p. 390)
• inverse function (p. 405) • radicand (p. 362) • square root function (p. 415)
• inverse relation (p. 405) • rational exponent (p. 382)
• like radicals (p. 374)
• nth root (p. 361)
Choose the correct term to complete each sentence.
1. The number under a radical sign is called the (index/radicand).
2. (Radical functions/Inverse functions) are of the form f (x) = 1n x.
3. A radical expression can always be rewritten using a(n) (rational exponent/inverse
relation).
4. When two functions are combined so the range of one becomes the domain of the
other, the resulting function is called a (square root function/composite function).
422 Chapter 6 Chapter Review
6-1 Roots and Radical Expressions
Quick Review Exercises
You can simplify a radical expression by finding the roots. Find each real root.
The principal root of a number with two real roots is the
positive root. The principal nth root of b is written as 1n b, 5. 125 6. 10.49
where b is the radicand and n is the index of the radical
expression. 7. 13 - 8 8. -13 8
For any real number a, 2n an = a if n nisiosdedven. Simplify each radical expression. Use absolute value
if symbols when needed.
e 0a0
9. 281x2 10. 23 64x6
Example 11. 24 16x12 12. 25 0.00032x5
What is the simplified form of 236x6? 13. 5 9x 4 14. 23 125x6y9
262x6 Find the root of the integer. 36
= 262(x3)2 Find the root of the variable.
= 6 0 x3 0 Take the square root of each term. Since the index
is even, include the absolute value symbol to
ensure that the root is positive even when x3 is
negative.
6-2 Multiplying and Dividing Radical Expressions
Quick Review Exercises
If 1n a and 1n b are real numbers, then 1n a # #Multiply if possible. Then simplify. #17. 12 18
1n b
1 1n a 21 1n b 2 = 1n ab, and, if b ≠ 0, then = na . 15. 13 9 13 3 16. 13 - 7 13 49
5b
To rationalize the denominator of an expression, rewrite it #Multiply and simplify.
so that the denominator contains no radical expressions. 18. 28x2 22x2 #19. 523 9y2 13 24y
Example Divide and simplify.
#What is the simplest form of 232x2y 218xy3? 23 81x5y3 24 162x4
13 3x2 14 2y8
2(32x2y)(18xy3) Combine terms. 20. 128 21. 22.
# #= 2142 2x2y2 132 2xy32 Factor. 5 8
# #= 242 32 22x3y4 Consolidate like terms. Divide. Rationalize all denominators.
# #= 242 32 22(x2x)(y2)2 23 6x2y4
Identify perfect squares. 23. 1168 24. 23x5 25. 2 13 5x7y
# #= 4 3 2xy2 1x = 24xy2 1x Extract perfect squares. 8x2
Chapter 6 Chapter Review 423
6-3 Binomial Radical Expressions
Quick Review Exercises
Like radicals have the same index and the same radicand. Add or subtract if possible.
Use the distributive property to add and subtract them. Use
the FOIL method to multiply binomial radical expressions. 26. 10 127 - 4 112
To rationalize a denominator that is a square root binomial, 27. 3 120x + 8 145x - 4 15x
multiply the numerator and denominator by the conjugate 28. 23 54x3 - 23 16x3
of the denominator.
Example Multiply.
What is the simplified form of 118 + 150 - 18? 29. 13 + 12 214 + 12 2
30. 1 15 + 11121 15 - 1112
118 + 150 - 18 31. 110 + 16 2110 - 13 2
# # #= 232 2 + 252 2 - 222 2 Factor.
= 3 12 + 5 12 - 2 12 Simplify each radical. Divide. Rationalize all denominators.
= (3 + 5 - 2) 12 Combine like terms. 32. 2 + 15 33. 3+ 118
15 1+ 18
= 6 12 Simplify.
6-4 Rational Exponents
Quick Review Exercises 1
You can rewrite a radical expression with a rational Simplify each expression. #35. 814
exponent. By definition, if the nth root of a is a real number 31
1
and m is an integer, then amn = 1n am = 1 1n a 2m; if m is 37. 52 52
# 34. 252
negative then a ≠ 0. Rational exponents can be used to 11
simplify radical expressions.
36. 163 43
Write each expression in simplest form.
Example ( ) 38. 1 4 39. 1 - 8y9 1
x4 23
Multiply and simplify 1x1 24 x32. 40. 1 29xy224 ( )41. 11 -18
#1x1 24 x32 1 3 x6 y3
= x2 x4 Rewrite with rational exponents. ( ) 42. 19
x3
5 x4 -15 43. ° y-23 ¢
= x4 Combine exponents. x-1
= 24 x5 Rewrite as a radical expression.
424 Chapter 6 Chapter Review
6-5 Solving Square Root and Other Radical Equations
Quick Review Exercises
To solve a radical equation, you must isolate a radical Solve each equation. Check for extraneous solutions.
expression on one side of the equation. You can then
rewrite the radical expression using a rational exponent and 44. 2 + 1x + 5 = 4 45. 3 12x + 6 = 18
use the reciprocal of the exponent to solve the equation.
1 2
For example, to solve a square root equation, you square
each side of the equation. Check all possible solutions in 46. 5(3x + 1)4 = 10 47. 4(3x - 3)3 = 36
the original equation to eliminate extraneous solutions.
48. 13x + 3 - 1 = x 49. 1x + 6 + 2 = x + 6
50. 15x + 1 - 2 1x = 1 51. 12x + 9 - 1x = 3
Example 52. Electricity The power P, in watts, that a circular
2
solar cell produces and the radius of the cell r in
What is the solution of 4(x - 2)3 = 16?
centimeters are related by the square root equation
2 Isolate the radical. r = 50.0P2p. About how much power is produced by
a cell with a radius of 12 cm?
(x - 2)3 = 4
23 3 3
((x - 2)3) 2 = 42 Raise both sides to the 2 power.
63 Law of exponents.
(x - 2)6 = 42
0 x - 2 0 = 8 Simplify.
x = 10 or x = - 6 Solve for x.
6-6 Function Operations Exercises
Quick Review Let f (x) = x − 4 and g (x) = x2 − 16. Perform each
function operation and then find the domain.
When performing function operations, you can use the
same rules you used for real numbers, but you must take # 53. f(x) + g(x) 54. g (x) - f (x)
into consideration the domain and range of each function. g(x)
The composition of function g with function f is defined as 55. f (x) g (x) 56. f (x)
(g ∘ f )(x) = g ( f (x)).
Let g (x) = 5x − 2 and h (x) = x2 + 1. Find the value of
Example each expression.
Let f (x) = x + 3 and g (x) = x2 - 2. What is (g ∘ f )( -2)? 57. (h ∘ g)( -1) 58. (h ∘ g)(0)
g( f ( - 2)) = g(( - 2) + 3) Evaluate f(- 2).
= g(1) Simplify. 59. (g ∘ h)(2) 60. (g ∘ h)(a)
= (1)2 - 2 Evaluate g(f(- 2)).
= - 1 Simplify. 61. Discounts A grocery store is offering a 50% discount
off a $4.00 box of cereal. You also have a $1.00
Therefore, (g ∘ f )( -2) = -1. off coupon for the same cereal. Use a composite
function to show whether it is better to use the
coupon before or after the store discount.
Chapter 6 Chapter Review 425
6-7 Inverse Relations and Functions
Quick Review Exercises
If a relation or a function is described by an equation in x Find the inverse of each function. Determine whether
and y, you can interchange x and y to get the inverse. The each inverse is a function.
domain of a function becomes the range of its inverse, and
the range of a function becomes the domain of its inverse. 62. f (x) = 2x2 - 8 63. f (x) = 15 - 3x
64. f (x) = 1x + 6 65. f (x) = (2x - 3)2
Example Graph each function and its inverse. Describe the domain
and range of each.
What is the inverse of f (x) = 1x - 10?
y = 1x - 10 Rewrite using y. 66. f (x) = 4x - 1 67. f (x) = (x + 3)2
Interchange the x and y values.
x = 1y - 10 Square each side. 68. f (x) = 1x - 3 69. f (x) = 6 - 5x2
x2 = y - 10 Solve for y.
y = x2 + 10 Write the inverse function. 70. Geometry The volume of a cube is determined
f -1(x) = x2 + 10 by the formula V = s3, where s is the length of one
side. Find the inverse formula. Use it to find the side
The domain of f (x) is x Ú 10, which means the range of length of a cube with a volume of 64 ft3.
f -1(x) is y Ú 10. Also, since the range of f (x) is y Ú 0, the
domain of f -1(x) is x Ú 0.
6-8 Graphing Radical Functions
Quick Review Exercises
The function f (x) = 1x is the parent function of the graph each function. Find the domain and range.
square root function f (x) = a1x - h + k. The graph
of f (x) = a1x is a stretch (a 7 1) or a shrink (0 6 a 6 1) 71. y = 1x - 5 72. y = 1x + 8
of the parent function. The graph of f (x) = a1x - h + k is
a translation h units horizontally and k units vertically 73. y = 5 1x + 9 74. y = - 1x - 4
of y = a1x. The graph of f (x) = 1n x is transformed by a, h,
and k in the same way as the graph of f (x) = 1x. 75. y = 23 x + 10 76. y = - 23 x - 2 + 5
Example Rewrite each function to make it easy to graph using
transformations. Describe each graph.
Describe the graph of y = 14x + 12.
77. y = 19x - 27 + 4 78. y = -3 14x - 16
79. y = 23 8x + 24
80. y = x - 4 + 6
5 4
y = 14x + 12
y = 14(x + 3) Factor the polynomial. Solve each equation by graphing.
y = 2 1x + 3 Simplify the radical. 81. 5 = - 1x - 3
82. 18x - 16 = 2 1x + 2
The graph of y = 14x + 12 is the graph of y = 2 1x
translated 3 units to the left.
426 Chapter 6 Chapter Review
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