2 Solution Key • Chapter 7, page 190 CHECK YOUR READINESS page 372 1. (3y - 2)(y - 4) = 3y2 - 12y - 2y + 8 = 3y2 - 14y + 8 2. (7a + 10)(7a - 10) = 49a2 - 70a + 70a …

Algebra 2 Solution Key • Chapter 7, page 190

CHECK YOUR READINESS page 372

1. (3y - 2)(y - 4) = 3y2- 12y - 2y + 8 =

3y2- 14y + 8 2. (7a + 10)(7a - 10) =

49a2- 70a + 70a - 100 = 49a2

- 1003. (x - 3)(x + 6)(x + 1) =(x - 3)(x2

+ 7x + 6) = x3+ 7x2

+ 6x -

3x2- 21x - 18 = x3

+ 4x2- 15x - 18

4. (3x3)2= 32(x3)2

= 9x6 5. (2b-2)(4b5) = 8b3

6. (xy-3)2= x2(y-3)2

= x2y-6=

7. = 6a2a4= 6a6 8. =

9. x2- 5x - 14 = 0; (x - 7)(x + 2) = 0;

x - 7 = 0 or x + 2 = 0; x = 7 or x = -210. 2x2

- 11x + 15 = 0; (2x - 5)(x - 3) = 0;

2x - 5 = 0 or x - 3 = 0; x = or x = 3 11. 3x2+

10x - 8 = 0; (3x - 2)(x + 4) = 0; 3x - 2 = 0 or x +

4 = 0; x = or x = -4 12. 12x2- 12x + 3 = 0;

4x2- 4x + 1 = 0; (2x - 1)2

= 0; 2x - 1 = 0; x =

13. 8x2- 98 = 0; 4x2

- 49 = 0; (2x - 7)(2x + 7) = 0;

2x - 7 = 0 or 2x + 7 = 0; x = or x = -

14. x4- 14x2

+ 49 = 0; (x2- 7)2

= 0; x2- 7 = 0;

x2= 7; x = 4 � 42.65 15. domain {1, 2, 3, 4},

range {2, 3, 4, 5} 16. domain {1, 2, 3, 4}, range {2}17. domain {all real numbers}, range {all real numbers} 18. domain {all real numbers}, range {all real numbers � 3}19. y = 2x2

- 4 20. y = -3(x2+ 1)

21. y = (x - 3)2+ 1 22. y = -(x + 4)2

- 512

!7

72

72

12

23

52

23ab6

4ab23

6a2b318a2

3a24

x2

y6

23. y = (x + 2)2- 1 24. y = 7 - (5 - x)2

ALGEBRA I REVIEW page 374

1. (3a2)(4a6) = 3(4)a2+ 6= 12a8

2. (-4x2)(-2x-2) = -4(-2)x2- 2= 8x0

= 8(1) = 83. (4x3y5)2

= 42(x3)2(y5)2= 16x6y10 4. (2x-5y4)3

=

23(x-5)3(y4)3= 8x-15y12

= 5. = 4a5- 2=

4a3 6. = 2x7- (-1)y5= 2x8y5 7. =

8. = = 9. (-6m2n2)(3mn) =

-6(3)m2+ 1n2+ 1= -18m3n3 10. (3x4y5)-3

=

= = 11. =

= = 12. x5(2x)3= x5(23)(x3) =

8x5+ 3= 8x8 13. = x4- 2- (-5)

= x7

14. = = = 18y5

15. (4p2q)(p2q3) = 4p2+2q1+3= 4p4q4 16. =

2x3- 1= 2x2 17. (p2)-2

= p-4= 18. =

-5x4-1= -5x3 19. = r2-2 s3-4 t4- (-4)

=

r0 s-1 t8= 20. ? = = 3x2

21. (s2t)3(st) = (s2)3(t3)(st) = s6 t3st = s6+ 1 t3+1=

s7t4 22. (3x-3y-2)-2= 3-2(x-3)-2(y-2)-2

=

x6y4= 23. (h4k5)0

= 1 24. ? =

r3- 1 s2+ 1 t3- 1= r2s3t2 25. Answers may vary.

Sample: 150= 1; = 45- 3

= 42= 16; (2 ? 5)2

=

22 ? 52= 4 ? 25 = 100; = ; 23 ? 25

= 23+ 5=

28; 4-3= = ; (32)3

= 32 ?3= 36

= 729. In (ab)m=

164

143

25

35Q23R5

45

43

sr3

ts2t3

rx6y4

9132

6x2y2

2y26xy2

xy2

2t8

s

r2s3t4

r2s4t24

215x4

3x1

p4

4x3

2x

144x4y12

8x4y7122(x2)2(y6)2

8x4y7(12x2y6)2

8x4y7

x4x22

x25

r8s5

r23s5

222r2s24t0

2rs

(2r21s2t0)22

2rs1

27x12y151

33(x4)3(y5)31

(3x4y5)3

9x4

432(x2)2

22Q3x2

2 R2

12xy5

(4x2)0

2xy56x7y5

3x21

8a5

2a28y12

x15

14

Chapter

7Radical Functions and Rational Exponents pages 372–435

�2 21Ox

y

2

�4

�2 2Oxy

�12

2 4Ox

y12

4

�2�4�6 O xy

�4

�8

�2�6 Ox

y

2

4 6Ox

y

6

4

2

ambm and = , the exponent is distributed to

both factors and to both the numerator and thedenominator.

7-1 Roots and Radical Expressionspages 375–379

Check Skills You’ll Need For complete solutions seeDaily Skills Check and Lesson Quiz Transparencies orPresentation CD-ROM. 1. 52 2. 0.32

3. 4. (x5)2 5. (x2y)2 6. (13x3y6)2

CA Standards Check pp. 376–377 1a. Since 05= 0,

the fifth root of 0 is 0; since (-1)5= -1, the fifth root

of -1 is -1; since 25= 32, the fifth root of 32 is 2.

1b. Since 0.012= 0.0001 and (-0.01)2

= 0.0001, thesquare roots of 0.0001 are 0.01 and -0.01; -1 has no

real square root; since = and = ,

the square roots of are and . 2a. =

= -3 2b. = = 3 2c. =

= 7 3a. = = =

2 Δx« y2 3b. = = =

-3c2 3c. = = =

x2Δy3

« 4a. w = ; 3 = ; d3= 12; d = � 2.29;

2.29 in. 4b. w = ; 5.5 = ; d3= 22; d = �

2.80; 2.80 in. 4c. w = ; 6.25 = ; d3= 25;

d = � 2.92; 2.92 in.

Exercises pp. 378–379 1. 152= 225 and (-15)2

=

225; square roots of 225 are 15 and -15 2. (0.07)2=

0.0049 and (-0.07)2= 0.0049; square roots of 0.0049

are 0.07 and -0.07 3. has no real square roots

4. = and = ; square roots of

are and 5. (-4)3= -64; the cube root of -64

is -4 6. (0.5)3= 0.125; the cube root of 0.125 is 0.5

7. = ; the cube root of is , or

8. (0.07)3= 0.000343; the cube root of 0.000343 is 0.07

9. 24= 16 and (-2)4

= 16; the fourth roots of 16 are 2and -2 10. -16 has no real fourth roots 11. 0.34

=

0.0081 and (-0.3)4= 0.0081; the fourth roots of 0.0081

are 0.3 and -0.3 12. = and =

; the fourth roots of are and

13. = = 6 14. = - = -6

15. ; no real root 16. = = 0.6

17. - = - = -4 18. = =

-4 19. - = - = -3 20. ; no real

root 21. = = = 4Δx«

22. = = = 0.5Δx3«"(0.5x3)2"0.52(x3)2"0.25x6

"(4x)2"42x2"16x2

"4 281"4 34"4 81

"3 (24)3!3264"3 43!3 64

"0.62!0.36!236

"622!36"62!36

2103

103

10,00081

10,00081

Q2103 R

410,00081Q10

3 R4

21223

62 272162 27

216Q236R

3

2 813

813

64169

64169Q2 8

13R264

169Q 813R

2

2 1121

!3 25

d3

4d3

4

!3 22d3

4d3

4

!3 12d3

4d3

4

"4 (x2y3)4"4 (x2)4(y3)4"4 x8y12

"3 (23c2)3"3 (23)3(c2)3"3 227c6

"(2xy2)2"22x2(y2)2"4x2y4"72

!49"4 34!4 81"3 (23)3

!32272 611

611

36121

36121Q2

611R

236121Q 6

11R2

Q27R2

am

bmQabRm

23. = = = x4Δy9

«

24. = = = 8b24

25. = = = -4a

26. = = = 3y2

27. = = = x2Δy3

«

28. = = = 2y2

29. V = pr3; 3V = 4pr3; r3= ; r = ; r =

� 1.34; 1.34 in. 30. r = � 1.68; 1.68 ft

31. r = � 0.48; 0.48 cm 32. r = �

0.08; 0.08 mm 33. x2= 100; 102

= 100 and (-10)2=

100, so x = 10,-10 34. x4= 1; 14

= 1 and (-1)4= 1,

so x = 1,-1 35. x2= 0.25; 0.52

= 0.25 and (-0.5)2=

0.25, so x = 0.5, -0.5 36. x4= ; = and

= , so x = ,- 37. = = -4;

- = - = -(-4) = 4; = = 8;

= = 2; from least to greatest: , ,

- , 38a. K = 1.35 ; 8 = 1.35 ; =

; L = � 35; about 35 ft 38b. K = 1.35 ;

10 = 1.35 ; = ; L = � 55; 55 - 35 =

20; about 20 ft longer 39. = = 0.5

40. = = = 41. =

= 0.2 42. = = 43. =

= 2Δc« 44. = =

3xy2 45. = =

12y2z2ΔxΔ 46. = = y4

47. = = -y4 48. =

= k3 49. = = -k3

50. = Δx + 3« 51. =

= (x + 1)2 52. = Δx«

53. = = x2 54. = =

Δx3« 55. Answers may vary. Sample: ,

- , 56a. true for all positive integers56b. true for all odd positive integers 57. yes,

because 10 is really 101 58. = x2, true for all

values of x, because x2 is always positive 59. =

x3, true for some values of x: x � 0 60. = x2,

true for some values of x: x = -1, 0, 1 61. =

Δx«, true for some values of x: x � 0 62. = Δm«

63. = = m2 64. =

= Δm3« 65. = = m4

66. = m 67. = = m2"n (m2)n"n m2n"n mn

"n (m4)n"n m4n"n (m3)n

"n m3n"n (m2)n"n m2n

!n mn

"3 x3

"3 x8"x6

"x4

"5 232x10"4 16x8

"3 28x6

"2n(x3)2n"2n

x6n"2n(x2)2n"2n

x4n

"2nx2n"f(x 1 1)2g2

"(x 1 1)4"(x 1 3)2

"5 (2k3)5"5 2k15"5 (k3)5

"5 k15"5 (2y4)5"5 2y20

"5 (y4)5"5 y20!xz

"xz(12xy2z2)2"144x3y4z5!3 3

"3 3(3xy2)3"3 81x3y6"4 (2c)4

"4 16c414Å

4 Q14R4

Å4 1

256"4 (0.2)4

!4 0.001613

26Å

3 Q26R3

Å3 8

216

"3 0.53!3 0.125

Q 101.35R

2101.35!L!L

!LQ 81.35R

281.35

!L!L!L!64!3264

!6 64!3264"6 26"6 64

"82!64"3 (24)3!3264

"3 (24)3!326423

23

1681Q22

3R4

1681Q23R

41681

Å3 3(0.002)

4pÅ3 3(0.45)

4p

Å3 3(20)

4pÅ3 3(10)

4p

Å3 3V

4p3V4p

43

"(2y2)5"5 25(y2)5"5 32y10

"4 (x2y3)4"4 (x2)4(y3)4"4 x8y12

"3 (3y2)3"3 33(y2)3"3 27y6

"(24a)3"(24)3a3"3 264a3

"(8b24)2"82(b24)2"64b48

"(x4y9)2"(x4)2(y9)2"x8y18


68. = = m3 69. =

= m4 70. = = x3 is true for only

some values of x since the expression is always positive and the expression x3 is sometimes positive andsometimes negative; the answer is B. 71. The realcomponent of point A is located on the horizontal axis;the imaginary component of point A is located on thevertical axis; the real component is 6; the imaginarycomponent is -5i; the point is (6, -5i); the complexnumber is 6 - 5i; the answer is B. 72. Use the formulaV = lwh; V = (x + 1)(x - 2)(x - 5) = (x2

- x - 2) (x - 5) = x3

- 5x2- x2

+ 5x - 2x + 10 = x3- 6x2

+

3x + 10; the answer is D. 73. (x + y)5= x5

+ 5x4y+ 10x3y2

+ 10x2y3+ 5xy4

+ y5 74. (2 - 3y)4= 24

+

4(2)3(-3y) + 6(2)2(-3y)2+ 4(2)(-3y)3

+ (-3y)4=

16 - 96y + 216y2- 216y3

+ 81y4 75. (3x - 5)6=

(3x)6+ 6(3x)5(-5) + 15(3x)4(-5)2

+ 20(3x)3(-5)3+

15(3x)2(-5)4+ 6(3x)(-5)5

+ (-5)6= 729x6

-

7290x5+ 30,375x4

- 67,500x3+ 84,375x2

- 56,250x +

15,625 76. (2a - b)7= (2a)7

+ 7(2a)6(-b) +21(2a)5(-b)2

+ 35(2a)4(-b)3+ 35(2a)3(-b)4

+

21(2a)2(-b)5+ 7(2a)(-b)6

+ (-b)7= 128a7

-

448a6b + 672a5b2- 560a4b3

+ 280a3b4- 84a2b5

+

14ab6- b7 77. y = 4x3

- 49x = x(4x2- 49) =

x(2x - 7)(2x + 7) 78. y = 81x2+ 36x + 4 =

(9x)2+ 2(9x)(2) + 22

= (9x + 2)2 79. y = 4x3+

8x2+ 4x = 4x(x2

+ 2x + 1) = 4x(x + 1)2

80. y = 12x3+ 14x2

+ 2x = 2x(6x2+ 7x + 1) =

2x(6x + 1)(x + 1) 81. y = 3x2- 7 = 3(x - 0)2

- 782. y = -2x2

+ x - 10 = - 10 =

-2 x2- x + - 10 + =

83. y = + 2x - 1 = (x2+ 8x) - 1 =

(x2+ 8x + 16) - 1 - 4 = (x + 4)2

- 5

7-2 Multiplying and Dividing RadicalExpressions pages 380– 385

Check Skills You’ll Need For complete solutions seeDaily Skills Check and Lesson Quiz Transparencies orPresentation CD-ROM. 1. 6 2. 3 3. 34. x2 5. ab 6. 5a3b4

CA Standards Check pp. 380–383 1a. ? =

= = 6 1b. ? = =

= = -3 1c. ? , not possible

2. = = ? =

5x2 ; = = ? =

x 3. 3 ? 2 = 3 ?

2 = 6 =

6 = 6(7x3y) = 42x3y 4a. =

= = 3 4b. = = ="4x3Å

12x4

3x"12x4

!3x!9Å

24327

!243!27

!3!3"3 ? (7x3y)2

"147(x3)2y2"7 ? x3 ? 21 ? x3 ? y2

"21x3y2"7x3!3 18x

!3 18x "3 x3"3 18 ? x ? x3"3 18x4!2

!2"(5x2)2"2 ? 52 ? (x2)2"50x4!4 24!4 4"3 (23)3!3227

!3 3(29)!329!3 3!36!3 ? 12

!12!3

14

14

14

x2

4

2 79822Qx 2 1

4R21

81

16R12Q

22Qx2 2 12xR

"4 x12x

124"4 x12"n (m4)n

"n m4n"n (m3)n"n m3n= 2x 4c. = =

= = 4x3 5a. =

? = 5b. = = ?

= 5c. = = =

= 6. a = ; at2 = d; t2 = ;

t = = ? =

Exercises pp. 383–385 1. =

2.

3.

4. 5. ,

not possible 6. = 5

7. 8.

9.

10.

11.

12. = = ?

= 13. = =

? 14. =

= ? =

15. =

= ? =

16. = =

? = 17. ? =

= = = 18. ?

= = = ?

= 19. ? = =

= ? =

20. ? = =

= 21. ? =

= =

22. ? = =

= 23. = =

= 10 24. = = =

25. = = = 2x2y2

26. = = = 5x"3 x2y2"3 125x5y2

Å3

250x7y3

2x2y"3 250x7y3

"3 2x2y

!2"8x4y4Å

56x5y5

7xy"56x5y5

!7xy

4xyÅ

16x2

y2Å48x3

3xy2"48x3

"3xy2!100

Å50050

!500!50

22x2y!3 30x22"3 30(x2)3 ? xy3

22"3 30x7y32"3 15x5y2"3 2x2y2

30y2!3 2y6"3 53 ? 2 ? (y2)3 ? y6"3 250y7

2"3 50y43"3 5y340xy!320"22 ? 3 ? x2y2

20"12x2y25"6xy24!2x7x3y4!6y

!6y"(7x3y4)2"72 ? 6 ? (x3)2 ? (y4)2 ? y

"294x6y9"42xy9"7x58y3!5y!5y

"(8y3)2"82 ? 5 ? (y3)2 ? y"320y7"40y2

"8y52!3 12"3 23 ? 12!3 96!3 6 ? 16

!3 16!3 62y "4 4x3y2"4 4x3y2"4 (2y)4

"4 24 ? 4 ? x3 ? y4 ? y2"4 64x3y625x2y"3 2y2

"3 2y2"3 (25x2)3"3 (25)3 ? 2 ? (x2)3 ? y3 ? y2

"3 2250x6y510a3b3!2b

!2b"(10a3b3)2"102 ? 2 ? (a3)2 ? (b3)2 ? b

"200a6b73y3!3 2y!3 2y 5"3 (3y3)3

"3 33 ? 2 ? (y3)3 ? y"3 54y102a"3 4a2"3 4a2

"3 (2a)3"3 23 ? 22 ? a3 ? a2"3 32a55x2!2x

5 "(5x2)2 ? !2x 5"50x5 5 "52 ? 2 ? (x2)2 ? x

3"3 3x2"3 33 ? "3 3x2 5"3 81x2 5 "3 33 ? 3 ? x2 5

2x!5x"22 ? 5 ? x2 ? x 5 "(2x)2 ? !5x 5

"20x3 5!3212(218) 5 !3 216 5 6

!3218 5!3212 ?!3 9 ? !3224 5 !32216 5 26

!325 ? !3225 5 !3 125

!25 ? !5!4 32 5 !4 8 ? 32 5 !4 256 5 4!4 8 ?

!3 9 ? !3281 5 !3 9(281) 5 !32729 5 29

!3 4 ? !3 16 5 !3 4 ? 16 5 !3 64 5 4!256 5 16

!8 ? !32 5 !8 ? 32

!daa

!a!a

!d!aÅ

da

da

dt2

"3 18x2

3xÅ3 2 ? (3x)2

3x ? (3x)2

Å3 2

3xÅ3 4

6x!3 4!3 6x

x!5y5y

!5y!5y

"x2

!5yÅ2x3

10xy"2x3

!10xy!35

5!5!5

!7!5

Å75"4 x2"4 44 ? (x3)4 ? x2"4 256x14

Å4 1024x15

4x"4 1024x15

!4 4x!x"22x2 ? x


27. = ? = 28. = ? =

= 29. = ? =

30. = ? 31. =

? 32.

33.

34.

35. F = ;

Fr2= Gm1m2;

36a.

36b. ?

36c. Answers may vary. Sample: First simplify

the denominator. Since = = , to rationalize the denominator, multiply the fraction by

. This yields 37. ?

= = = 10 38. ? =

= 39.

3x6y5

40.

20x2y3 41. ( + 7) =

+ = 10 + 42. 3(5 + ) =

15 + 3 43. ( + ) = + =

5 + 5 44. ? ? = =

= 45.

46.

47.

48.

49.

50.

51. 232 Å

1112x ?

!3x!3x

5Å11x3y12x4y

53"11x3y

22"12x4y5 23

2

"3 2xy2

xy!3 14"3 7x2y

?"3 72xy2

"3 72xy2 5"3 73 ? 2xy2

"3 73x3y3 57"3 2xy2

7xy 5

10"3 5x2 ?

"3 52x"3 52x

5 10!3 25x5x 5 2!3 25x

x"3 3x2

3x

"3 3x2

"3 33x3 51!3 9x

? "3 3x2

"3 3x2 5!7x!7x

5 5!14x3(7x) 5 5!14x

21x

5!23!7x

?!2y!2y

5"10x4y

"4x2y4 5x2!10y

2xy2 5x!10y

2y2

"5x4

"2x2y3 ?"3 27x7 5 "3 33(x2)3x 5 3x2!3 x

"3 3x2 ? "3 x2 ? "3 9x3 52x!3 2"3 23 ? 2 ? x3

"3 16x3"3 2x2!3 4!3 2x!3

!75!25!15!5!5!21

!217!27!2!100

!50!2!y10"22(x2)2(y3)2y 5

5"2xy6 ? 2"2x3y 5 10"4x4y7 5

!2y3"2x12y11 5 3"2(x6)2(y5)2y 5

"x5y5 ? 3"2x7y6 5"3 43 ? 5 5 4!3 5!3 320

!3 80!3 4!2"102 ? 2!200!40

!5!2 ? 2 1 !3 ? 27!2 ? 2

5 2 1 !614 .!2

!2

7!2!2 ? 49!98

!6 1 315

!3!3

5!3(!2 1 !3)

!2255!2 1 !3

!75!6 1 3

15

5!6 1 1575 5!150 1 !225

75 5!75(!2 1 !3)

75 5

!2 1 !3!75

? !75!75

5!Gm1m2F

F

!Gm1m2!F

? !F!F

5

r 5 ÅG1m2

F5r2 5

Gm1m2F

;

Gm1m2

r2x!5y!2

? !2!2

5 x!102yÅ

5x2

2y2 5

Å5x4y2x2y3 5

"5x4y

"2x2y3 5!5y!5y

5!15y

5y!3!5y

?

Å3

5y 5Å3xy2

5xy3 5"3xy2

"5xy3 55x2!55"5x4 5

5Å60x5

12x 515"60x5

3!12x5"4 53

"4 53 5!4 250

5!4 2!4 5

!4 2!4 5

"3 (3x)2

"3 (3x)2 5"3 45x2

3x!3 5!3 3xÅ

3 53x

!3 4x2

"3 22

"3 22!3 x!3 2

!3 x!3 2

!10x4x

!10x"16x2

!2x!2x

!5!8x

!5!8x

!2x2

!2!2

!x!2

!x!2


52. -2( + )= -

53.

54.

55. altitude of 100 mi: v = � 17,498;

altitude of 200 mi: v = � 17,286;

17,498 - 17,286 = 212; about 212 mi/h greater

56.

57. A product of two square roots can be

simplified in this way only if the square roots are real

numbers; and are not. 58. h = 16t2 =

16( )2= 16(18a5) = 288a5; 288a5 ft

59. For some values; it is easy to see that the equation

is true if x = 0 or x = 1. But when x � 0, is not a

real number, although is. 60. Check students’

work. 61. 2xy

62. = 8x3y6; = 2xy2

63. = 20; 64. =

= 65.

66. = =

67.

-a = 2c; a = -2c;

-b = 6d; b = -6d 68. No changes

need to be made; since they are both odd roots, there is no need for absolute value symbols. 69. Use Pascal’striangle to expand (a + b)6; (a + b)6

= a6+ 6a5b +

15a4b2+ 20a3b3

+ 15a2b4+ 6ab5

+ b6; substitute 2a fora and -3b for b into each term; A: 6a5b = 6(2a)5(-3b) =6(32a5)(-3b) = -576a5b; B: 15a4b2

= 15(2a)4(-3b)2=

15(16a4)(9b2) = 2160a4b2; the answer is B.70. The graph of y = 3(x + 3)2

- 2 is a vertical shift 2units down of the graph y = 3(x + 3)2; the answer is A.71. Line 1 has a slope of 1 and a y-intercept of (0, 0);the equation of line 1 is y = x; line 2 has a slope ofand a y-intercept of (0, 3); the equation of line 2 is y = ; write the equations in slope-interceptform; line 1 is y = x and line 2 is 3x + 5y = 15; theanswer is C. 72. Solve the quadratic equation x2

+

3x - 28 = 0 by factoring. x2+ 3x - 28 = 0; (x - 4)

(x + 7) = 0; x = 4 and x = -7; The roots are 4 and -7.The answer is B.

73. = = 11Δa45Δ

74. - = - = -9c24d32

75. = = -4a27

76. = = 2y5 77. ="0.25x6"5 25(y5)5"5 32y25"3 (24)3(a27)3"3 264a81

"92(c24)2(d32)2"81c48d64"112 ? (a45)2"121a90

2 35x 1 3

235

1y6d; 1

y2b 51

y6d;

"yb 5 1y3d; yb 51

x2c; 1x2a 5

1x2c;

"xa 5 1xc; xa 5"6 x4

"6 y3 ?"6 y3

"6 y3 5"6 x4y3

y

Å6 x4

y3Å6

y23

x24Å5

yx4 5

!5 y

"5 x4 ?!5 x!5 x

5!5 xy

x

"5 x24y 51"3 xy2 ?

"3 x2y

"3 x2y5"3 x2y

xyÅ31

xy2

"3 x21y22!20 5 2!5!3 8000

"3 8x3y6"64x6y12

4x2y2; "4x2y2 5"16x4y4 5

"3 x2"x3

"18a5!28!22

20!22 cm25 !8800 5 "202 ? 22 5 20!22; !440 (!20)

Å1.24 3 1012

3950 1 200

Å1.24 3 1012

3950 1 100

!3 2 !2!8

? !2!2

5 !6 2 !4!16

5 !6 2 24

3!5 1 55

3 1 !5!5

? !5!5

55 24!3 4 2 6!3 22"3 33 ? 2

22"3 23 ? 4!3 54!3 3223!33x12x 5 2!33x

4x

= 0.5Δx3Δ 78. =

= x2y5 79 =

= 2Δx9Δy24

80. = = 0.08x20

81. y2- 4y + 16

y3+ 4y2

-4y2+ 0y

-4y2- 16y

16y - 6416y + 64

-128quotient: y2

- 4y + 16, R -128; not a factor

82. x2- 3x + 9

x3+ 3x2

-3x2+ 0x

-3x2- 9x

9x + 279x + 27

0quotient: x2

- 3x + 9; factor

83.

quotient: 3a2- a, R 4; not a factor

84.

quotient: 2x3+ x2

+ 2x, R 10; not a factor

85. 86.

87. 88.

89. 90.

91. 92.

7-3 Binomial Radical Expressions pages 386–390

Check Skills You’ll Need For complete solutions seeDaily Skills Check and Lesson Quiz Transparencies orPresentation CD-ROM. 1. 15x2

+ 2x - 82. -24x2

+ 71x - 35 3. x2- 16 4. 16x2

- 255. x2

+ 10x + 25 6. 4x2- 36x + 81

Q 35(2) R

25 9

100Q2 34(2) R

25 9

64

Q0.32 R

25 (0.15)2 5 0.0225Q2 1

3(2) R25 1

36

Q2112 R2 5 121

4Q112 R

25 121

4

Q2102 R2 5 (25)2 5 25Q10

2 R25 52 5 25

x3 1 0x2

x3 2 2x2

2x2 2 4x 2x2 2 4x 0 1 10

2x3 1 x2 1 2xx 2 2 q 2x4 2 3x3 1 0x2 2 4x 1 10 2x4 2 4x3

22a2 2 a22a2 2 a 0 1 4

3a2 2 a2a 1 1 q 6a3 1 a2 2 a 1 4 6a3 1 3a2

x 1 3qx3 1 0x2 1 0x 1 27

y 1 4qy3 1 0y2 1 0y 2 64

"(0.08)2(x20)2"0.0064x40

"4 24(x9)4(y24)4"4 16x36y96"7 (x2)7(y5)7

"7 x14y35"(0.5)2(x3)2 CA Standards Check pp. 386–388 1a. + =

(2 + 3) = 1b. - ; cannot

combine 1c. + = (4 + 5) =

2. height = 2(6 ) = ; length = 3( ) =18 ; perimeter = 2( + ) = 2(30 ) =60 , or about 84.9 in. 3. + -

5 = + - = +

- = (5 + 12 - 15) =

4. ( - )( - ) = - 2( )( ) +

( ) = 2 - 2 + 3 = 5 - 25. ( + )( - ) = ( )2

- ( )2=

5 - 2 = 3 6. ? =

Exercises pp. 388–390 1. + = (5 + 1) =

= 2. - = (6 - 2) =

3. + ; cannot combine 4. - =

(3 - 5) = -2 5. 14 + ; cannot

combine 6. - = (7 - 2) =

7. + = + = +

= (18 + 15) = 8. -

= - = - =

(28 - 15) 9. + = +

= 3 + 4 = (3 + 4) = 7

10. + = + = 3 +

2 = (3 + 2) = 5 11. 3 - 2 =

3 - 2 = 9 - 6 12. +

= + = 2 + 213. (3 + )(1 + ) = 3 + 3 + 1 + 5 =

8 + 4 14. (2 + )(1 + 3 ) = 2 + 6 +

1 + 21 = 23 + 7 15. (3 - 4 )(5 - 6 ) =15 - 18 - 20 + 48 = 63 - 3816. ( + )2

= ( + )( + ) =3 + 2 + 5 = 8 + 2 17. ( + 6)2

=

( + 6)( + 6) = 13 + 12 + 36 =

49 + 12 18. (2 + 3 )2=

(2 + 3 )(2 + 3 ) = 20 + 12 + 18 =

38 + 12 19. (5 - )(5 + ) =25 - 11 = 14 20. (4 - 2 )(4 + 2 ) =16 - 12 = 4 21. (2 + 8)(2 - 8) = 24 - 64 =

-40 22. ( + )( - ) = 3 - 5 = -223. ? = = =

-2 + 2 24. ? = =

25. ? = =

13 + 7 26. ? = =

= =

27. + + = 6 + 4 + 3 =

(6 + 4 + 3) = 13 28. + 2 - 5 =

5 + 8 - 5 = 8 29. 5 + 4 =!98x!32x!3!3!3!3!3!48!75!2!2

!2!2!2!18!32!72

11 1 8!2214

22 1 16!2228

22 1 8!8228

6 1 8!8 1 164 2 32

2 1 2!82 1 2!8

3 1 !82 2 2!8

!3

10 1 7!3 1 34 2 3

2 1 !32 1 !3

5 1 !32 2 !3

12!3 1 823

12!3 1 827 2 4

3!3 1 23!3 1 2

43!3 2 2

!3

4 2 4!322

4 2 4!31 2 3

1 2 !31 2 !3

41 1 !3

!5!3!5!3!6!6

!3!3!11!11!10!10!2!5!2!5

!2!5!13!13!13!13!13!15!15

!5!3!5!3!5!3!2!2!2

!2!2!7!7!7!7!7!5

!5!5!5!5!4 3!4 2"4 24 ? 3"4 24 ? 2!4 48

!4 32!3 2!3 3"3 33 ? 2"3 33 ? 3

!3 54!3 81!3 2!3 2!3 2

!3 2"3 23 ? 2"3 33 ? 2!3 16!3 54

!2!2!2!2"42 ? 2

"32 ? 2!32!1813!5!5 5

15!528!53"52 ? 514"22 ? 53!125

14!2033!2!215!2

18!23"52 ? 26"32 ? 23!506!18

5"3 x2"3 x22"3 x27"3 x23!y!x!x!x

5!x3!x4!3 34!3

4!3 3!3 32!3 36!3 36!6!6

!6!65!6

39 1 10!1524 1 6!15 1 4!15 1 1516 2 15 5

4 1 !154 1 !15

6 1 !154 2 !15

!2!5!2!5!2!5!6!6!3!3

!3!2!2(!2)!3!2!3!22!2!215!212!2

5!25"32 ? 23"42 ? 2"52 ? 2!18

3!32!50!2!218!212!2!2

6!212!2!29!xy!xy5!xy4!xy

2!3 57!4 55!7!7

3!72!7


20 + 28 = (20 + 28) = 4830. - 4 + 2 = 5 - 12 + 8 =

5 + (-12 + 8) = 5 - 4 31. 4 +

3 = 4(6y) + 3(3y) = 24y + 9y =

(24y + 9y) = 33y 32. 3 - 4 +

= 3(2) - 4(3) + 8 =

(6 - 12 + 8) = -233. ( - )( + 2 ) = 3 + 2 -

- 14 = -11 +

34. (2 + 3 )(5 - 7 ) = 50 - 14 +

15 - 42 = 8 +

35. (1 + )(5 + ) = 5 + + 5 +

= 5 + + 30 + 12 = 17 + 3136. (2 - )(3 + ) = 6 + 2 - 3 -

= 6 + 6 - 21 - 42 = -36 - 1537. ( + )( + 2 ) = x + 2 +

+ 6 = x + 3 + 6 38. (2 - 3 )(4 -

5 ) = 8y - 10 - 12 + 30 = 8y -

22 + 30 39. = ? =

= 40. ?

= =

= =

- 41. ? =

=

= =

42. ? = =

43. ? = =

44. ? = = =

- 45. Golden Ratio = reciprocal

of Golden Ratio = = =

= = = ;

the difference between the Golden Ratio and its

reciprocal = -

= = = 1 46. +

= + possibilities for a: 2(12), 2(22),2(32), . . . ; a must be twice a perfect square.47. Answers may vary. Sample: Without simplifyingfirst, you must estimate three separate square roots, andthen add the estimates. If they are first simplified, thenthey can be combined as 13 Then only one square !2.

!a;6!2

!a!7222

1 1 !5 1 1 2 !52

1 1 !5 2 (21) 2 !52

21 1 !52 ;1 1 !5

2

21 1 !52

1 2 !522

2(1 2 !5)24

2(1 2 !5)1 2 5

2(1 2 !5)(1 1 !5)(1 2 !5)

21 1 !5

1 1 !52 ;!3 122!3 2

8!3 2 2 4!3 124

4!3 16 2 2!3 964

"3 42

"3 424 2 2!3 6!3 4

x 1 5"4 x3

x5"4 x3 1 x

x"4 x3

"4 x35 1 !4 x!4 x

2 1 3!3 42

3!3 4 1 22

"3 22

"3 223 1 !3 2!32

!3 2 !72

22!3 1 4!724

5!3 1 5!7 2 3!7 2 7!324

5!3 1 5!7 2 !63 2 !1473 2 7

!3 1 !7!3 1 !7

5 2 !21!3 2 !7

!22!3

22!3 1 !221

4!2 2 4!3 1 2!3 2 3!221

4!2 2 4!3 1 !12 2 !182 2 3

!2 2 !3!2 2 !3

4 1 !6!2 1 !3

89 1 42!32239

8 1 36!3 1 6!3 1 814 2 243

2 1 9!32 1 9!3

4 1 3!32 2 9!3

4 1 !272 2 3!27

!2y

!2y!2y!2!y!2!y!3x!3x

!3x!3!x!3!x!2!2!2!1764

!98!18!18!98!2!2!2!144

!72!2!2!72!10!10

!10!2!5!2!5!21!21

!21!7!3!7!3!3 2!3 2

!3 2!3 2!3 2!3 128

!3 54!3 16!6!6!6!6!6!6"54y2

"216y2!2!3!2!3!2!2!3!32!18!75

!2x!2x!2x!2x root need be estimated. 48. d = rt; t = =

= = = , or

about 4.53 s 49. Answers may vary. Sample: ( + 2)( )(2 + )(2 - ) 50. A = /w;A = (3 + )x(1 + 2 )y = (3x + x)(y + 2 y)= 3xy + 6 xy + xy + 10xy = 13xy + 7 xy =

(13 + 7 )xy = the answer is D.

51. + = +

= = = -

52. - = -

= =

= 4 53. (a = 0 and b � 0) or (b = 0 and a � 0) 54. In the second step the exponentwas incorrectly distributed: (a - b)x 2 ax

- bx.55a. m and n can be any positive integers. 55b. m mustbe even or n must be odd. 55c. m must be even, and ncan be any positive integer. 56. = = |x| is truefor all values of x since the expression is alwayspositive and the expression |x| is always positive; theanswer is A. 57. (8 - 5i)2

= (8 - 5i)(8 - 5i) = 64 -80i + 25i2 = 64 - 80i + 25(-1) = 64 - 80i - 25 = 39- 80i; the answer is B. 58. Find factors with product4(-4) = -16 and sum 15: 16 and -1; 4x2

+ 16x - x -4; 4x(x + 4) - 1(x + 4); (x + 4)(4x - 1); the answer is D.59. The x-coordinate of the vertex = = =

-8; the y-coordinate of the vertex is y = -(-8)2-

16(-8) - 62, y = 2; the vertex is (-8, 2); the vertex liesin the second quadrant; the answer is C.

60. ? = = = 361. ? = = x

62. = = = 4 63. = =

= = 64. ? = =

= 2x 65. ? = = 7x2

66. = = ? =

67. ? = 68. 2x3- 16 =

0; x3- 8 = 0; x3

- 23= 0; (x - 2)(x2

+ 2x + 4) = 0;

x = 2 or x = = =

= -1 69. x3+ 1000 = 0; x3

+

103= 0; (x + 10)(x2

- 10x + 100) = 0; x = -10 or

x = = =10 4 !2300

22(210) 4 "(210)2 2 4(1)(100)

2(1)

4 i!322 4 2i!32

22 4 !2122

22 4 "22 2 4(1)(4)2(1)

"3 100x2

5x"3 (5x)2

"3 (5x)2!3 4!3 5xÅ

3 45x 5

!3nn

!n!n

!3!nÅ

6m2mn 5 Å

3n

!6m!2mn

!2"98x4"14x3!7x"3 (2x)3

"3 8x3!3 4x"3 2x2!933

2!936

!3726

!62 ? !6!6 ? !6

!62!6

!16Å322

!32!2

!15"15x2!5x!3x!3 2"3 33 ? 2!3 54!3 18!3 3

2 S 2162 ? 21 T2 b

2a

"4 x4x

44"4 x4

!38!32

4!5 1 4!3 2 4!5 1 4!35 2 3

4(!5 2 !3)(!5 1 !3)(!5 2 !3)

4(!5 1 !3)(!5 2 !3)(!5 1 !3)

4!5 1 !3

4!5 2 !3

12

224

1 1 !5 1 1 2 !51 2 5

1(1 2 !5)(1 1 !5)(1 2 !5)

1(1 1 !5)(1 2 !5)(1 1 !5)

11 1 !5

11 2 !5

!5!5!5!5

!5!5!5!5!5!2!5!2!7 2 2

!7

60 2 20!27

60 2 20!29 2 2

20(3 2 !2)(3 1 !2)(3 2 !2)

203 1 !2

dr


= 5 70. 125x3- 1 = 0; (5x)3

-

13= 0; (5x - 1)(25x2

+ 5x + 1) = 0; 5x - 1 = 0 or

25x2+ 5x + 1 = 0; x = or x = =

= = 71. x4-

14x2+ 49 = 0; (x2

- 7)2= 0; x2

- 7 = 0; x2= 7;

x = 72. 25x4- 40x2

+ 16 = 0; (5x2)2-

2(5x2)(4) + 42= 0; (5x2

- 4)2= 0; 5x2

- 4 = 0;

5x2= 4; x2

= x = = = ? =

73. 81x4- 1 = 0; (9x2

- 1)(9x2+ 1) = 0;

9x2- 1 = 0 or 9x2

+ 1 = 0; x2= or x2

= -

x = or x =

CHECKPOINT QUIZ 1 page 390

1. = = 2. =

= xy2 3. = = -a

4. = = -y2 5. =

+ 3(8) = + 24 6. ? =

= 12x3y 7. - =

- 8. ? = =

= 9. =

5 - 12 = -7 10. ? = =

7-4 Rational Exponents pages 391–396

Check Skills You’ll Need For complete solutions seeDaily Skills Check and Lesson Quiz Transparencies orPresentation CD-ROM.

1. 2. 3. 4. 5. 6.

CA Standards Check pp. 391–394 1a. = =

= 2 1b. ? = = 2

1c. ? = = = 4 2a. = =

; z0.4= = = 2b. = ;

= 2c. If m is negative, then = ,

and if a = 0, then the denominator of the fractionwould be zero. Since this cannot happen, a 2 0.

3. N = � 0.270 revolutions per second, or

about 16 revolutions per minute 4a. =

= = 4b. = = = 823(!5 32)332351

125153

1(!25)3

25232

4.90.5

2p(1.7)0.5

1(!n a)2ma

mny

32(!y)3

x23"3 x2"5 z2z

25z

4101

"8 y3

1y

38

y238!16!2 ? !88

122

12

!2 ? !22122

12"4 24

!4 161614

b2

16a616b12

a812

1125x6y3

19x2

116

20!7 2 2587

20!7 2 25112 2 25

4!7 2 54!7 2 5

54!7 1 5

(!5 2 2!3)(!5 1 2!3)4!10x45x

4y!10x45xy

4"10xy2

9(5xy)!5y!5y

4!2xy

9"5x2y9!3 212!3 3

3!3 544!3 81!356"140x6y2

3"28x3y22"5x38!3!192

!8(!24 1 3!8)"5 (2y2)5"5 2y10

"3 (2a)3"3 2a3"5 (x)5(y2)5

"5 x5y10u b uc2"4 (b)4(c2)4"4 b4c8

4i34

13

19;1

9

42!5

5

!5!5

42!5

42!5

4Å45

45;

4!7

21 4 i!310

25 4 5i!350

25 4 !27550

25 4 "52 2 4(25)(1)2(25)

15

4 5i!310 4 10i!32

4c. = = = 16 5. =

? = ? =

Exercises pp. 394–396 1. = = = 6

2. = = = 3 3. = = = 74. ? = ? = 10 5. ? ?

= ? ? = = -3

6. ? = ? = = = 6

7. ? = ? = = = 8

8. ? = ? = = = = 3

9. ? = ? = = =

= 3 10. = 11. = 12. =

or 13. = or 14. =

or 15. = or 16. =

= or 17. = = or

18. = 19. = or

20. = = 21. =

= 22. = = 23. =

= 24. = = 25. =

= (5xy)2 or 25x2y2 26. �

72.8; about 72.8 m 27. � 15.1;

about 15.1 m 28. � 7.9; about 7.9 m

29. � 1.6; about 1.6 m

30. = = 22= 4 31. = = 42

=

16 32. = = (-2)2= 4 33. =

= (-2)6= 64 34. = = =

35. = = = 23= 8 36. =

= = 43= 64 37. 10,0000.75

= =

= 103= 1000

38. = = 39. = =

40. = 41. = 42. =

= 43. = =

44. = = = 45. =

= or = =

46. = = 47. =

= 48. = = x3y9

49. = = 50. = =

= -7 51. = = =

-3 52. 321.2= = = 26

= 64(!5 32)63265

"5 (23)5!52243(2243)15"3 (27)3

!32343(2343)13

y5

x10x210

y25ax223

y213b

15

x3

y29a x14

y234b

12y2

x8x28y2

Ax23y2

16B212y4

x3x23y4Ax12y2

23B26

x4!3 x"3 (x4)3 ? x"3 x13x133(x13)

13

Q x2

x211R131

xx21(x4)214Q x3

x21R21

4

22y3(232)15y3(232y15)

152 3

x322713x23

(227x29)135

x23

5Ax23B211

3x23

A3x23B21

1x4x24Ax24

7B71x2x22Ax2

3B23

(!4 10,000)3

10,00034(!16)316

32

161.5(!4)343241.51

16

124

1(!5 32)4322

45(!5232)6

(232)65(!328)2(28)

23

(!3 64)264 23(!3 8)28

23

h 50.00252(25)2.27

2.3

h 50.00252(50)2.27

2.3

h 50.00252(50)2.27

1.2

h 50.00252(100)2.27

1.2f(5xy)6g13

"3 (5xy)6c12(c2)

14"4 c2a

23Aa1

3B2(!3 a)2a

23(a2)

13"3 a2(7x)

32A7x

12B3

(!7x)3(7x)32(7x3)

12"(7x)3

x327

12(7x3)

12"7x3(210)

12!210

(!5 y)6"5 y6y65y1.2(!x)3"x3x

32

x1.51(!4 t)3

1"4 t3t2

341

(!8 y)91"8 y9

y298(!5 y)2"5 y2y

25(!7 x)2"7 x2

x27!5 xx

15!6 xx

16"4 34

!4 81!4 3 ? 27!4 27!4 327143

14

"3 33!3 27!3 3 ? 9!3 9!3 39133

13

!64!2 ? 32!32!232122

12

!36!3 ? 12!12!312123

12

(!323)3!323!323!323(23)13

(23)13(23)

13!10!1010

1210

12

"72!494912"3 33!3 2727

13

"62!363612

12x5

1x5

1!3 8

x258213

(8x15)213(22)4(!5232)4(232)

45


53. 2431.2= = = 36

= 72954. 643.5

= = = 87= 2,097,152

55. 1004.5= = = 109 or 1,000,000,000

56. 32-0.4= = = = 57. 64-0.5

=

= = 58. = =

= 59. = = 2(2)3= 16

60. = = =

61. = = = 101= 10

62. = � 78%; = � 61%;

= � 37% 63. � 636;

the answer is A. 64. = = =

6(2)7= 768 65. ? = = = =

66. ? = = = = 67. �

= = = = 68. � = =

= = 69. = =

70. = = =

71. = = 72. =

= 73. � = ? � =

= = = 74. ?

= = = =

75. = = = 76. -1=

-= = 77. The cube root of -64

is -4, which equals . The square root of -64 is

not a real number, but = -8.

78. The exponent applies only to the 5, not to the 25.

79. Answers may vary. Sample: Let a = 4 - , then

a = = 16 - 5 = 11, which is

rational; a = , a = a = ; no

80a. ? ? ? = x ? x = x2, so =

80b. = = = =

81. = 72= 49 82. =

= 32= 9 83. = x4p-2p

= x2p

84. ? = ? = =

50= 1 85. = = = =

86. = = 34- 2= 32

= 93(21!2)(22!2)(321!2)22!2

3!232!2

232!2(32)

1!29

1!2

52!322!3(52)2!352!3252!352!3

x4p

x2p331!52(11!5)

331!5

311!5(7!2)!2!x

x12x

24(x2)

14"4 x2!x

"4 x2!x!x!x!x

4 2 512

22Q4 1 512R,4 2 5

12

Q4 1 512RQ4 2 5

12RQ4 1 5

12R

512

12

2(64)12 5 2!64

2(64)13

1xy

12

y212x2

12

13Ax3

2y32B

C("x3 y3)13D1

x13

x213(x21)

13C Ax21

2B2D13

1x

724

x2724x

18242

21242

424x

342

782

16x2

16

Ax34 4 x

78B1

x1336

x21336x

6361

5362

2436x

161

5362

23

x23x

536x

16x

23Ax1

2 ? x5

12B139y8

4x681

12y8

1612x6

a81y16

16x12b124x7

9y916

12x7

8112y9

a16x14

81y18b12

1x

14y

56

x214 y2

56x

122

34 y2

132

12

x12 y2

13

x34 y

12

x16y

14y2

141

12x

232

12

x23 y2

14

x12 y2

12

y12y

714y

10142

314

y572

314y

314y

57x

12x

510x

6102

110x

352

110

x110x

35y

45y

810y

5101

310y

121

310y

310y

12

x12x

714x

4141

314x

271

314x

314x

27

6(!5 32)76(32)75PV

75

0.036(46 3 104)34(2.7)2

80008033R

A

(2.7)240008033R

A(2.7)220008033R

A

104

103(!3 1000)4

(!100)31000

43

10032

2 181

21(23)4

21(!3227)42(227)2

43

2(!4 16)32(16)341

361

(26)2

1(!32216)2(2216)2

231

81!64

64212

14

122

1(!5 32)2322

25

(!100)910092

(!64)76472

(!5 243)624365 87. -20 = 35.74 + 0.6215(5) - 35.75V0.16

+

0.4275(5)V0.16; -20 = 38.8475 - 35.75V0.16+

2.1375V0.16- 58.8475 = -33.6125V0.16; V0.16

=

; = ; V � 33.13; 33.13 mph

88. = = = = n�5;

the answer is D. 89. a-m= is true for all real

number values of a, m, and n; a-m= represents a

property of exponents; the answer is B.90. The parent function is y = |x|; the graph of y - 2 =2|x - 2| or y = 2|x - 2| + 2, is translated 2 units right,2 units up and is stretched by a factor of 2; the answer is C. 91. - = (6 - 2) =

92. + = + =

+ = (9 + 12) =

93. = = 1 +

94. = =

+ 7 = 95. =

= =

96. =

= =

= 97. 4x3- 8x2

+

16x = 4x(x2- 2x + 4) 98. x2

+ 4x + 4 = (x + 2)2

99. x2- 18x + 81 = (x - 9)2 100. 16a2

- 9b2=

(4a)2- (3b)2

= (4a - 3b)(4a + 3b) 101. 25x2-

40xy + 16y2= (5x)2

- 2(5x)(4y) + (4y)2= (5x - 4y)2

102. 9x2+ 48x + 64 = (3x)2

+ 2(3x)(8) + 82=

(3x + 8)2

7-5 Solving Square Root and OtherRadical Equations pages 397–402

Check Skills You’ll Need For complete solutions seeDaily Skills Check and Lesson Quiz Transparencies orPresentation CD-ROM. 1. -3, 2 2. -2, 7

3. 1, 4. , 2 5. , 6. ,

CA Standards Check pp. 397–399 1. -6= 0; = 6; 5x + 1 = 36; 5x = 35; x = 7

2. = 54; = 27; = ;

x + 3 = 9; x = 6 3. r = ; 20 = ; 400

= ; P = 400(0.02p); P = 25.1; about 25.1 watts; it isabout 4 times the power of a 10-cm cell.4. + 3 = x; = x - 3; 5x - 1 =

(x - 3)2; 5x - 1 = x2- 6x + 9; x2

- 11x + 10 = 0;(x - 10)(x - 1) = 0; x - 10 = 0 or x - 1 = 0; x = 10or x = 1; check x = 10: + 3 0 10, +

3 0 10, 7 + 3 = 10; check x = 1: + 3 0 1,+ 3 0 1, 2 + 3 2 1; the only solution is 10.!4

!5(1) 2 1!49!5(10) 2 1

!5x 2 1!5x 2 1

P0.02p

% P0.02p% P

0.02p

2723C(x 1 3)

32D 23(x 1 3)

322(x 1 3)

32

!5x 1 1!5x 1 1

3222

31225

221323

2

10 2 8!27

6 2 2!2 2 6!2 1 47

6 2 2!2 2 3!8 1 !169 2 2

(22 1 !8)(23 1 !2)(23 2 !2)(23 1 !2)

22 1 !823 2 !2

4 1 6!5 1 2!10 1 15!2241

4 1 6!5 1 2!10 1 3!504 2 45

(2 1 !10)(2 1 3!5)(2 2 3!5)(2 1 3!5)

2 1 !102 2 3!5

15 2 4!148 2 2!56

(!8 2 !7)(!8 2 !7)(!8 2 !7)23!5

1!5 2 45 1 4!5 2(!5 2 1)(!5 1 4)

21!2!212!29!2

2"62 ? 23"32 ? 22!723!18

4!3 3!3 32!3 36!3 3

1a

m

1a

m

n53 ?2

31an

53b23

an(322(21

6))b23An3

2 4 n216B23

Q258.8475233.6125R

10016(V0.16)

10016258.8475

233.6125


5. - = 0; = ;3x + 2 = 2x + 7; x = 5; check x = 5: -

0 0, - 0 0, 0 = 0; the solution is 5.

Activity p. 400 1a. x = + 5; x - 5 =

; (x - 5)2= x + 7; x2

- 10x + 25 = x + 7;x2

- 11x + 18 = 0; (x - 9)(x - 2) = 0; x - 9 = 0 orx - 2 = 0; x = 9 or x = 2; 2 solutions 1b. 1 point ofintersection 1c. Yes, 2 is an extraneous solution1d. check x = 2: 2 0 + 5, 2 0 3 + 5, 2 2 8;check x = 9: 9 0 + 5, 9 0 4 + 5, 9 = 9; yes2a. 1 point of intersection, so 1 solution 2b. 1 pointof intersection, so 1 solution 2c. 2 points ofintersection, so 2 solutions

Exercises pp. 400–402 1. + 3 = 15; = 12;= 4; x = 42

= 16 2. - 1 = 3; = 4;

= 1; x = 12= 1 3. = 5; x + 3 = 52;

x = 25 - 3 = 22 4. = 4; 3x + 4 = 42;3x + 4 = 16; 3x = 12; x = 4 5. - 7 = 0;

= 7; 2x + 3 = 72; 2x + 3 = 49; 2x = 46; x =

23 6. - 2 = 0; = 2; 6 - 3x = 22;

6 - 3x = 4; 3x = 2; 7. = 4; x + 5 =

; x + 5 = 48; x = 3 or x = -13 8. = 9;x - 2 = ; x - 2 = 427; x = 29 or x = -25

9. = 24; = 8; x - 2 = ; x - 2 =

16; x = 18 10. = 81; = 27; x + 3

= ; x + 3 = 81; x = 78 11. - 2 = 25;

= 27; x + 1 = ; x + 1 = 9; x = 8

12. 3 + = 11; = 8; 4 - x = ;

4 - x = 4; x = 0 13. V = d3; 15,000 = d3;

d3= ; d = � 30.6; about 30.6 ft

14. Q = ? = ; v = ?

= ; A = pr2; Q = Av; 86,400 = pr2(7200);

r2= = ; r = � 1.95; d = 2r � 3.9; 4 in.

15. - 2x = 0; = 2x; 11x + 3 =

(2x)2; 11x + 3 = 4x2; 4x2- 11x - 3 = 0;

(4x + 1)(x - 3) = 0; 4x + 1 = 0 or x - 3 = 0;

x = (extraneous) or x = 3 16. - 3x = 0;

= 3x; 5x + 4 = (3x)2; 5x + 4 = 9x2;

9x2- 5x - 4 = 0; (9x + 4)(x - 1) = 0; 9x + 4 = 0

or x - 1 = 0; x = (extraneous) or x = 1

17. - 5 = x; = x + 5; 3x +

13 = (x + 5)2; 3x + 13 = x2+ 10x + 25; x2

+ 7x +

12 = 0; (x + 3)(x + 4) = 0; x + 3 = 0 or x + 4 = 0;x = -3 or x = -4 18. + 5 = x; =

x - 5; x + 7 = (x - 5)2; x + 7 = x2- 10x + 25;

x2- 11x + 18 = 0; (x - 2)(x - 9) = 0; x - 2 = 0

or x - 9 = 0; x = 2 (extraneous) or x = 9

19. - 1 = x; = x + 1; x + 3 =(x 1 3)12(x 1 3)

12

!x 1 7!x 1 7

!3x 1 13!3x 1 13

249

(5x 1 4)12

(5x 1 4)1221

4

!11x 1 3!11x 1 3Å

12p

12p

86,4007200p

7200 in.min

12 in.1 ft

600 ftmin

86,400 in.3minQ12 in.

ft R350 ft3

min

Q90,000p R

13

15,000(6)p

p6

p6

823(4 2 x)

32(4 2 x)

32

2723(x 1 1)

32

(x 1 1)3227

43

(x 1 3)343(x 1 3)

34

843(x 2 2)

343(x 2 2)

34

932

(x 2 2)234

32

(x 1 5)23x 5 2

3

!6 2 3x!6 2 3x!2x 1 3

!2x 1 3!3x 1 4!x 1 3!x

4!x4!x!x3!x3!x

!9 1 7!2 1 7

!x 1 7

!x 1 7

!17!17!2(5) 1 7!3(5) 1 2

!2x 1 7!3x 1 2!2x 1 7!3x 1 2 (x + 1)2; x + 3 = x2+ 2x + 1; x2

+ x - 2 = 0;(x + 2)(x - 1) = 0; x = -2 (extraneous) or x = 1

20. = x + 1; 5 - x = (x + 1)2; 5 - x =

x2+ 2x + 1; x2

+ 3x - 4 = 0; (x + 4)(x - 1) = 0;

x = -4 (extraneous) or x = 1 21. = ;

3x = x + 6; 2x = 6; x = 3 22. - =

0; = ; = ;

(x + 5)2= 5 - 2x; x2

+ 10x + 25 = 5 - 2x; x2+

12x + 20 = 0; (x + 10)(x + 2) = 0; x + 10 = 0 or x + 2 = 0; x = -10 (extraneous) or x = -2

23. = ; 7x + 6 = 9 + 4x; 3x = 3;

x = 1 24. = x - 1; 3x + 7 = (x - 1)2;

3x + 7 = x2- 2x + 1; x2

- 5x - 6 = 0;(x + 1)(x - 6) = 0; x + 1 = 0 or x - 6 = 0;x = -1 (extraneous) or x = 6 25. - x = 1;

= x + 1; x + 7 = (x + 1)2; x + 7 = x2+

2x + 1; x2+ x - 6 = 0; (x + 3)(x - 2) = 0;

x + 3 = 0 or x - 2 = 0; x = -3 (extraneous) or x = 226. = x + 3; -3x - 5 = (x + 3)2;-3x - 5 = x2

+ 6x + 9; x2+ 9x + 14 = 0;

(x + 7)(x + 2) = 0; x + 7 = 0 or x + 2 = 0; x = -7

(extraneous) or x = -2 27. - =

0; = ; 3x + 2 = 2x + 7; x = 5

28. x + 8 = ; (x + 8)2= x2

+ 16;

x2+ 16x + 64 = x2

+ 16; 16x = -48; x = -3

29. = ; 2x = x + 5; x = 5

30. 1 = ; 12= 3 + x; x = 1 - 3 = -2

31a. y1 = - x; 31b. y1 = ;y2 = 1 y2 = x + 1

31c. y1 = - x - 1;y2 = 0

31d. The graph of each pair consistsof two straight lines, one of which ishorizontal. They intersect at differentpoints, but these points have thesame x-value, about 1.236.

Ox

y

3

1

1 3

!5

!5!5

(3 1 x)12

(x 1 5)12(2x)

12

Ax2 1 16B12!2x 1 7!3x 1 2

(2x 1 7)12(3x 1 2)

12

!23x 2 5

!x 1 7!x 1 7

!3x 1 7

(9 1 4x)12(7x 1 6)

12

C(5 2 2x)14D4C(x 1 5)

12D4(5 2 2x)

14(x 1 5)

12

(5 2 2x)14(x 1 5)

12

!x 1 6!3x

(5 2 x)12


2 4Ox

y4

2

Ox

y4

2 4

32a. ; 2A = ;

? = ; s = =

32b. � 8.8; about 8.8 in. 32c. �

8.8 � 15.2; about 15.2 in. 33. A = s2; s = ; s == 4 , or about 5.7 m; the answer is B.

34. 3 - 3 = 9; 3 = 12; =

4; 2x = 42; 2x = 16; x = 8 35. + 1 = 5;

= 4; = 2; 2x = 23; 2x = 8; x = 436. - 3 = 0; = 3; 2x - 1 = 32;2x - 1 = 9; 2x = 10; x = 5 37. + 7 = 0;

= -7; 2x + 3 = (-7)2; 2x + 3 = 49;2x = 46; x = 23 (extraneous); no real solution

38. = x + 1; x2+ 3 = (x + 1)2; x2

+ 3 =

x2+ 2x + 1; 2 = 2x; x = 1 39. - 3 = 5;

= 8; 2x + 3 = ; 2x + 3 = 16; 2x = 13;x = 6.5 40. + 4 = 36; = 32;

= 16; x - 1 = ; x - 1 = -8 or x - 1 = 8;x = -7 or x = 9 41. - = 2; - 2 =

; ( - 2)2= ( )2; x - 4 + 4 =

x - 5; -4 = -9; (-4 )2= (-9)2; 16x = 81;

x = 42. = + 2; - 2 = ;

= x - 8; x + 4 + 4 = x - 8; 4 =

-12; = -3; x = (-3)2= 9

43. The solution is x = 2.

Check: -

= 0 ✓

44. The solution is x = -1and x = -6.Check: +

= 5 ✓;

+

= 5 ✓


Check: =

✓8f3(2) 1 2g212

f3(2) 1 2g12

X Y1 Y21.7 2.6646 3.00231.8 2.7203 2.94091.9 2.7749 2.8832 2.8284 2.82842.1 2.881 2.77682.2 2.9326 2.7282.3 2.9833 2.6816

X =

X Y1 Y2-6.3 4.9731 5-6.2 4.9825 5-6.1 4.9915 5-6 5 5-5.9 5.0081 5-5.8 5.0159 5-5.7 5.0232 5

X =

!3 2 (26)

!(26) 1 10

!3 2 (21)

!(21) 1 10

X Y1 Y2-1.3 5.0232 5-1.2 5.0159 5-1.1 5.0081 5-1 5 5-.9 4.9915 5-.8 4.9825 5-.7 4.9731 5

X =

!4(2) 1 3

!5(2) 1 1X Y1 Y21.7 -.0483 01.8 -.0315 01.9 -.0154 02 0 02.1 .01478 02.2 .02899 02.3 .04268 0

X =

!x!x!x(!x 1 2)2

!x 2 8!x!x 2 8!x8116

!x!x!x!x 2 5!x!x 2 5!x(x 2 5)

12x

12

1634(x 2 1)

43

2(x 2 1)432(x 2 1)

43

843(2x 1 3)

34

(2x 1 3)34

"x2 1 3

(2x 1 3)12

(2x 1 3)12

!2x 2 1!2x 2 1(2x)

132(2x)

13

2(2x)13

!2x!2x!2x!2!32

!A!3

s!3s 5"2(200)!3

3

"2!3A3Å

2!3A9

2A!39

!3!3

s2 5 2A3!3

3s2!3A 5 3s2!32


Check: =

3 ✓


Check: + 1

= ✓


Check: -

2 = 0 ✓

49. The solution is x = -1.

Check: =

✓

50. The solution is x = 1.25

or .

Check: [2(1.25) - 1) =

[(1.25) + 1) ✓

51. v = ; v2= 64d; d = 52. Answers may

vary. Sample: = 53. +

= ; ( + )2= ( ;

x + 1 + 2 + 2x = 5x + 3; 2 =

2x + 2; = x + 1; 2x(x + 1) = (x + 1)2;2x2

+ 2x = x2+ 2x + 1; x2

= 1; x = -1 (extraneous)

or x = 1 54. = ; = 2x;= x; 2x = x2; x2

- 2x = 0; x(x - 2) = 0; x = 0

or x - 2 = 0; x = 0 or x = 2 55. = 2;

= 22; = 4 - x; 2x = (4 - x)2; 2x = 16- 8x + x2; x2

- 10x + 16 = 0; (x - 8)(x - 2) = 0;x - 8 = 0 or x - 2 = 0; x = 8 (extraneous) or x = 2

56. = ; = x + 5; x + 25= (x + 5)2; x + 25 = x2

+ 10x + 25; x2+ 9x = 0;

x(x + 9) = 0; x + 9 = 0 or x = 0;x = -9 (extraneous) or x = 0 57. Plan 1: Use acalculator to evaluate + 2 and store the result asx. Evaluate and store the result as x. Continuethis procedure about seven times, until it becomes clearthat the values are approaching 2. Plan 2: The given

!x 1 2!2

!x 1 25!x 1 5"!x 1 25

!2xx 1 !2x

"x 1 !2x

!2xx 1 !2x!2x"x 1 !2x

!2x(x 1 1)

!2x(x 1 1)!2x(x 1 1)

!5x 1 3)2!2x!x 1 1!5x 1 3!2x

!x 1 1!3x 1 5!x 2 3

v2

64!64d

g16

g13

54

X Y1 Y2.95 .96549 1.11771.05 1.0323 1.12711.15 1.0914 1.13611.25 1.1447 1.14471.35 1.1935 1.1531.45 1.2386 1.16111.55 1.2806 1.1688

X =

f2 1 3(21)g13

f2(21) 1 1g13

X Y1 Y2-1.3 -1.17 -1.239-1.2 -1.119 -1.17-1.1 -1.063 -1.091-1 -1 -1-.9 -.9283 -.8879-.8 -.8434 -.7368-.7 -.7368 -.4642

X =

!5(10) 2 25

!10(10)X Y1 Y29.7 .1535 09.8 .10154 09.9 .05038 010 0 010.1 -.0496 010.2 -.0985 010.3 -.1467 0

X =

(25)12

f(25) 2 9g12

X Y1 Y224.7 4.9623 4.969924.8 4.9749 4.9824.9 4.9875 4.9925 5 525.1 5.0125 5.0125.2 5.0249 5.0225.3 5.0373 5.0299

X =

!(7) 2 5

!4(7) 2 10X Y1 Y26.7 4.0988 3.91156.8 4.1473 4.02496.9 4.1952 4.13527 4.2426 4.24267.1 4.2895 4.34747.2 4.3359 4.44977.3 4.3818 4.5497

X =


equation is equivalent to x = Solve thisequation to find that x = 2. 58a. A counterexample isa = 3, b = -3. 58b. A counterexample is a = -5,b = 3. 59. A: x2

+ 5x - 36 = 0; (x + 9)(x - 4) = 0;x + 9 = 0 or x - 4 = 0; x = -9 or x = 4; the answer isA. 60. (6a3b4)(-2a-3b2) = (6)(-2)a(3-3)b(4+2)

=

-12(1)b6= -12b6; the answer is C. 61. Congruent

figures have the same size and shape. Since y = 3x2- 1

is a vertical stretch of y = x2 transposed down 1 and y = x2

- 1 is y = x2 transposed down 1, their graphs are not congruent.The answer is B. 62. =

2=

2= 42

= 16 63. 251.5= = ( )3

= 53= 125

64. ? = (6 ? 12 = = = = 6

65. ? = (8 ? 40 = = = =

8 66. ? = (3 ? 18 = = = =

= 3 67. 81-0.25= = = =

68. 43.5= = )7

= 27= 128

69. 125 ? = = = =

= 52= 25 70. 32 ? = = = 2

71. ? = ? = ? =

72. 7P1 = = = = 7 73. 7P3 =

= = = 7 ? 6 ? 5 = 210

74. 5P3 = = = = 5 ? 4 ? 3 =

60 75. 8P4 = = = =

8 ? 7 ? 6 ? 5 = 1680 76. 4P4 = = = =

4 ? 3 ? 2 ? 1 = 24 77. 5C2 = = =

= = 10 78. 7C5 = = =

= = 21 79. 5C5 = = =

= 1 80. 6C5 = = = = 6

81. 7C1 = = = = 7 82. x2-

7x + 12 = 0; (x - 3)(x - 4) = 0; x - 3 = 0 or x - 4 = 0; x = 3 or x = 4 83. x2

- 8x + 15 = 0;(x - 3)(x - 5) = 0; x - 3 = 0 or x - 5 = 0; x = 3 orx = 5 84. x2

+ 9x + 20 = 0; (x + 5)(x + 4) = 0;x + 5 = 0 or x + 4 = 0; x = -5 or x = -4 85. 3x2

+

8x + 4 = 0; (3x + 2)(x + 2) = 0; 3x + 2 = 0 or x + 2 = 0; x = - or x = -2 86. 9x2

+ 15x +

4 = 0; (3x + 1)(3x + 4) = 0; 3x + 1 = 0 or 3x + 4 =

0; x = - or x = 87. 4x2+ 11x + 6 = 0;

(4x + 3)(x + 2) = 0; 4x + 3 = 0 or x + 2 = 0;x = - or x = -2

GEOMETRY REVIEW page 403

1–4. Estimates may vary. 1a. Let d = diameter ofcylinder, then d2

= e2+ e2

= 2e2, and d = e . Then !2

34

243

13

23

7 ? 6!1 ? 6!

7!1! 6!

7!1!(7 2 1)!

6 ? 5!5! ? 1

6!5! 1!

6!5!(6 2 5)!

5!5! ? 1

5!5! 0!

5!5!(5 2 5)!

7 ? 62 ? 1

7 ? 6 ? 5!5! ? 2!

7!5! 2!

7!5!(7 2 5)!

5 ? 42 ? 1

5 ? 4 ? 3!2! ? 3!

5!2! 3!

5!2!(5 2 2)!

4!1

4!0!

4!(4 2 4)!

8 ? 7 ? 6 ? 5 ? 4!4!

8!4!

8!(8 2 4)!

5 ? 4 ? 3 ? 2!2!

5!2!

5!(5 2 3)!

7 ? 6 ? 5 ? 4!4!

7!4!

7!(7 2 3)!

7 ? 6!6!

7!6!

7!(7 2 1)!

1106

1103

1103QÅ 1

100R31

(!100)30.01321002

32

3216

32!256

256212("3 53)2

(!3 125)2125231251 2

131252

13

(!4472

13

1"4 34

1!4 81

1810.25!3 2"3 33 ? 2

"3 33 ? 2!3 545413)

1318

133

13!5

"82 ? 5!32032012)

1240

128

12

!2"62 ? 2!727212)

1212

126

12

!252532("3 43)

(!3 64)6423

!2 1 x. r = radius of cylinder =

Height of cylinder = e. =

= = about 57% more

volume 1b. =

= =

= � 1.26; about 26% more

surface area 2. = =

= = � 1.57; about 57% more surface

area 3. Let s = side length of square, then d =

diagonal of square = diameter of circle = =

= . So, r = radius of circle = .

= = = � 1.57;

= = = �

1.11. The area of the circle exceeds the area of thesquare by about 57%. The circumference of the circleexceeds the perimeter of the square by about 11%.4. The small sphere has a diameter of e, and a radius of

. = = =

3 � 5.20; about 420% more volume

7-6 Function Operations pages 404–410

Check Skills You’ll Need For complete solutions seeDaily Skills Check and Lesson Quiz Transparencies orPresentation CD-ROM. 1. domain: {0, 2, 4};range: {-5, -3, -1} 2. domain: {-1, 0, 1}; range: {0}3. domain: all real numbers; range: all real numbers4. domain: all real numbers; range: all real numbers � 0 5. 10 6. 28

CA Standards Check pp. 404–406 1. f (x) = 5x2-

4x, g(x) = 5x + 1; (f + g)(x) = 5x2- 4x + 5x + 1 =

5x2+ x + 1, domain: all real numbers; (f - g)(x) =

5x2- 4x - (5x + 1) = 5x2

- 9x - 1, domain: all realnumbers 2. f(x) = 6x2

+ 7x - 5, g(x) = 2x - 1;(f ? g)(x) = (6x2

+ 7x - 5)(2x - 1) = 12x3+ 14x2

-

10x - 6x2- 7x + 5 = 12x3

+ 8x2- 17x + 5,

domain: all real numbers; = =

= 3x + 5, domain: all real numbers

except 3a. ( f + g)(x) = f(g(x)) = f(x2) = x2- 2;

( f + g)(-5) = (-5)2- 2 = 25 - 2 = 23 3b. No; the

12

(3x 1 5)(2x 2 1)2x 2 1

6x2 1 7x 2 52x 2 1QfgR(x)

!3

3e3!3 ? 88 ~ e

3

43pQ

e!32 R3

43pQ

e2R

3Volume of large sphereVolume of small sphere

e2

p!24

s ? p!24s

2pQs!22 R

4sCircumference of circle

Perimeter of square

p2

2ps2

4s2

pQs!22 R2

s2Area of circle

Area of square

s!22s!2"2s2

"s2 1 s2

p2

3pe2

6e2

4p ? 3e2

46e2

4pQe!32 R2

6e2Surface area of sphereSurface area of cube

p 1 p!26

pe2 1 p!2e2

6e2

4pe2

4 1 2!2pe2

26e2

2pQe!22 R2 1 2pQe!2

2 Re6e2

Surface area of cylinderSurface area of cube

p2 < 1.57;2pe3

4e3

pQe!22 R2e

e3

Volume of cylinderVolume of cube

e!22 .


order in which operations are performed changesbetween ( f + g) and (g + f ), which changes what eachcomposition equals. 4a. Let f(x) = 0.9x and g(x) =0.75x. Then (g + f )(x) = g(0.9x) = 0.75(0.9x).4b. ( f + g)(x) = f (0.75x) = 0.9(0.75x) 4c. It makesno difference.Exercises pp. 406–410 1. f(x) + g(x) = 3x + 5 +

x2= x2

+ 3x + 5 2. g(x) - f(x) = x2- (3x + 5) =

x2- 3x - 5 3. f(x) - g(x) = 3x + 5 - x2

= -x2+

3x + 5 4. f(x) ? g(x) = (3x + 5)(x2) = 3x3+ 5x2

5. = 6. = 7. ( f + g)(x) =

f(x) + g(x) = 3x + 5 + x2= x2

+ 3x + 58. ( f - g)(x) = f(x) - g(x) = 3x + 5 - x2

= -x2+

3x + 5 9. (g - f )(x) = g(x) - f(x) = x2- (3x + 5) =

x2- 3x - 5 10. ( f ? g)(x) = f(x) ? g(x) =

(3x + 5)(x2) = 3x3+ 5x2

11. = = 12. = =

13. f(x) + g(x) = 2x2+ x - 3 + x - 1 =

2x2+ 2x - 4; domain: all real numbers 14. g(x) -

f(x) = x - 1 - (2x2+ x - 3) = -2x2

+ 2; domain:all real numbers 15. f (x) - g(x) = 2x2

+ x - 3 -

(x - 1) = 2x2- 2; domain: all real numbers

16. f(x) ? g(x) = (2x2+ x - 3)(x - 1) = 2x3

- 2x2+

x2- x - 3x + 3 = 2x3

- x2- 4x + 3;

domain: all real numbers 17. = =

= 2x + 3; domain: all real numbers

except 1 18. = = =

; domain: all real numbers except 1 and

19. (f ? g)(x) = f(x) ? g(x) = 9x(3x) = 27x2, domain:

all real numbers; = = = 3; domain: all

real numbers except 0 20. (g ° f )(x) = g(f(x)) =g(2x) = 2x + 3; (g + f )(3) = 2(3) + 3 = 9;(g + f )(-2) = 2(-2) + 3 = -1 21. (g ° f )(x) =g(f(x)) = g(x2) = Δx2

+ 5Δ = x2+ 5; (g + f )(3) =

Δ32+ 5Δ = Δ14Δ = 14; (g + f )(-2) = Δ(-2)2

+ 5Δ =Δ9Δ = 9 22. (h ° g)(1) = h(g(1)) = h(2 ? 1) = h(2) =22

+ 4 = 8 23. (h + g)(-5) = h(g(-5)) = h(2(-5)) =h(-10) = (-10)2

+ 4 = 104 24. (h + g)(-2) =h(g(-2)) = h(2(-2)) = h(-4) = (-4)2

+ 4 = 2025. (g ° h)(-2) = g(h(-2)) = g((-2)2

+ 4) = g(8) =2(8) = 16 26. (g + h)(0) = g(h(0)) = g(02

+ 4) =g(4) = 2(4) = 8 27. (g + h)(-1) = g(h(-1)) =g((-1)2

+ 4) = g(5) = 2(5) = 10 28. (g + g)(3) =g(g(3)) = g(2 ? 3) = g(6) = 2 ? 6 = 1229. (h + h)(2) = h(h(2)) = h(22

+ 4) = h(8) =82

+ 4 = 68 30. (h + h)(-4) = h(h(-4)) =h((-4)2

+ 4) = h(20) = 202+ 4 = 404

31. (g + f )(-2) = g( f(-2)) = g((-2)2) = g(4) =4 - 3 = 1 32. ( f + g)(-2) = f(g(-2)) = f(-2 - 3) =f(-5) = (-5)2

= 25 33. (g + f )(0) = g(f(0)) =

9x3x

f(x)g(x)QfgR(x)

232

12x 1 3

x 2 1(2x 1 3)(x 2 1)

x 2 12x2 1 x 2 3

g(x)f(x)

(2x 1 3)(x 2 1)x 2 1

2x2 1 x 2 3x 2 1

f(x)g(x)

x2

3x 1 5

g(x)f(x)QgfR(x)3x 1 5

x2f(x)g(x)QfgR(x)

x2

3x 1 5g(x)f(x)

3x 1 5x2

f(x)g(x)

g(02) = g(0) = 0 - 3 = -3 34. (f + g)(0) = f(g(0)) =f(0 - 3) = f(-3) = (-3)2

= 9 35. (g + f)(3.5) =g(f(3.5)) = g((3.5)2) = g(12.25) = 12.25 - 3 = 9.2536. (f + g)(3.5) = f(g(3.5)) = f(3.5 - 3) = f(0.5) =

(0.5)2= 0.25 37. (f + g) = = =

= = = 6.25 38. (g ° f) = g =

= = = -2.75 39. (f + g)(c) =

f(g(c)) = f(c - 3) = (c - 3)2= c2

- 6c + 940. (g + f )(c) = g(f(c)) = g(c2) = c2

- 341. ( f ° g)(-a) = f (g(-a)) = f(-a - 3) = (-a - 3)2

=

a2+ 6a + 9 42. (g + f )(-a) = g(f (-a)) =

g((-a)2) = g(a2) = a2- 3 43a. f (x) = 0.9x

43b. g(x) = x - 2000 43c. g(f(18,000)) =g(0.9(18,000)) = g(16,200) = 16,200 - 2000 = 14,200;$14,200 43d. f(g(18,000)) = f(18,000 - 2000) =f(16,000) = 0.9(16,000) = 14,400; $14,40044a. (g + f )(x) = g( f(x)) = g(0.12x) = 9.14(0.12x) =1.0968x 44b. (g + f )(15) = 1.0968(15) = 16.452;16.452 pesos 45. f (x) + g(x) = 2x + 5 + x2

- 3x +

2 = x2- x + 7 46. 3f(x) - 2 = 3(2x + 5) - 2 =

6x + 15 - 2 = 6x + 13 47. g(x) - f(x) = x2- 3x +

2 - (2x + 5) = x2- 5x - 3 48. -2g(x) + f(x) =

-2(x2- 3x + 2) + 2x + 5 = -2x2

+ 6x - 4 + 2x +

5 = -2x2+ 8x + 1 49. f(x) - g(x) + 10 = 2x + 5 -

(x2- 3x + 2) + 10 = 2x + 5 - x2

+ 3x - 2 + 10 =

-x2+ 5x + 13 50. 4f(x) + 2g(x) = 4(2x + 5) +

2(x2- 3x + 2) = 8x + 20 + 2x2

- 6x + 4 = 2x2+

2x + 24 51. -f(x) + 4g(x) = -(3x2+ 2x - 8) +

4(x + 2) = -3x2- 2x + 8 + 4x + 8 = -3x2

+ 2x +

16; domain: all real numbers 52. f (x) - 2g(x) = 3x2+

2x - 8 - 2(x + 2) = 3x2+ 2x - 8 - 2x - 4 =

3x2- 12; domain: all real numbers 53. f(x) ? g(x) =

(3x2+ 2x - 8)(x + 2) = 3x3

+ 6x2+ 2x2

+ 4x - 8x -

16 = 3x3+ 8x2

- 4x - 16; domain: all real numbers54. -3f(x) ? g(x) = -3(3x2

+ 2x - 8)(x + 2) =(-9x2

- 6x + 24)(x + 2) = -9x3- 6x2

+ 24x -

18x2- 12x + 48 = -9x3

- 24x2+ 12x + 48; domain:

all real numbers 55. = =

= 3x - 4; domain: all real numbers

except -2 56. = =

= 5(3x - 4) = 15x - 20; domain:

all real numbers except -2 57. Answers may vary.Sample: (g + f )(3) means g(f(3)), so first evaluate f (3) = 2(3) = 6; then evaluate g(6) = 6 + 1 = 7.58. f(g(1)) = f(3(1) + 2) = f(5) = = 1

59. g( f (-4)) = g = g(-2) = 3(-2) + 2 = -4

60. f(g(0)) = f(3(0) + 2) = f(2) = = 0

61. g(f(2)) = g = g(0) = 3(0) + 2 = 2Q2 2 23 R

2 2 23

Q24 2 23 R

5 2 23

5(3x 2 4)(x 1 2)x 1 2

5(3x2 1 2x 2 8)x 1 2

5f(x)g(x)

(3x 2 4)(x 1 2)x 1 2

3x2 1 2x 2 8x 1 2

f(x)g(x)

14 2 3gQ14RgQQ12R

2RQfQ12RRQ12R

254Q52R

2fQ52R

fQ12 2 3RfQgQ12RRQ12R


62a. (A + r)(x) when x = 2: A(r(2)) = A(12.5(2)) =A(25) = p ? 252

= 625p � 1963; the area after 2seconds is about 1963 in.2 62b. A(r(4)) = A(12.5(4)) =A(50) = p ? 502

= 2500p � 7854; about 7854 in.2

63. f(g(x)) = f (x2) = 3(x2) = 3x2; g(f(x)) = g(3x) =(3x)2

= 9x2 64. f(g(x)) = f(x - 5) = x - 5 + 3 =

x - 2; g(f(x)) = g(x + 3) = x + 3 - 5 = x - 265. f(g(x)) = f(2x) = 3(2x)2

+ 2 = 12x2+ 2;

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

+ 2) = 6x2+ 4

66. f(g(x)) = f(2x - 3) = = =

x - 3; g(f(x)) = g = 2 - 3 = x -

3 - 3 = x - 6 67. f(g(x)) = f(4x) = -4x - 7;g(f(x)) = g(-x - 7) = 4(-x - 7) = -4x - 28

68. f(g(x)) = f(x2) = ; g( f (x)) = g =

= 69. Answers may vary.Sample: 69a. g(x) = 0.12x 69b. f(x) = 9.50x69c. (g + f )(x) = 1.14x; your savings will be $1.14 foreach hour you work. 70a. f(x) and g(x) 70b. 0, 15,30; 3, 28, 103 70c. (f + g)(x) = f(g(x)) = f(x2

+ 3) =3(x2

+ 3) = 3x2+ 9 70d. 3*A1^2 + 9, 9, 84, 309

70e. (g + f )(x) = g(f(x)) = g(3x) = (3x)2+ 3 =

9x2+ 3 70f. 9*A1^2 + 3, 3, 228, 903 71a. P(x) =

I(x) - C(x) = 5995x - (1000 + 700x) = 5295x - 100071b. P(30) = 5295(30) - 1000 = 157,850; $157,85072a. g(x) is the bonus earned when x is the amount ofsales over $5000. h(x) is the excess of x sales over $5000.72b. (g + h)(x) because you first need to find the excesssales over $5000 to calculate bonus.73. ( f + g)(x) = f(x) + g(x) Def. of Function

Addition= 3x - 2 + (x2

+ 1) Substitution= x2

+ 3x - 2 + 1 Commutative Prop.of Addition

= x2+ 3x - 1 Arithmetic

74. ( f - g)(x) = f(x) - g(x)) Def. of Function Subtraction

= 3x - 2 - (x2+ 1) Substitution

= 3x - 2 - x2- 1 Opposite of a Sum

Property= -x2

+ 3x - 2 - 1 Commutative Prop.of Addition

= -x2+ 3x - 3 Arithmetic

75. (f + g)(x) = f(g(x)) Def. of Composition of Functions

= f(x2+ 1) Substitution

= 3(x2+ 1) - 2 Substitution

= 3x2+ 3 - 2 Distributive Property

= 3x2+ 1 Arithmetic

76a. f(x) = x + 10; g(x) = 1.09x 76b. f(g(x)) meanseach grade is increased 9% before adding the 10-pointbonus; f(g(75)) = f(1.09(75)) = f(81.75) = 81.75 +

10 = 91.75 76c. g( f(x)) means the 10-point bonus isadded, then the sum is increased by 9%; g( f(75)) =g(75 + 10) = g(85) = 1.09(85) = 92.65 76d. no

77. f(x) ? g(x)

x2 1 10x 1 254Qx 1 5

2 R2Qx 1 5

2 Rx2 1 52

Qx 2 32 RQx 2 3

2 R

2x 2 62

2x 2 3 2 32

= (x3- 3x2

- 5x + 15)= x7

- 3x6- 5x5

+ 15x4

+ 2x6- 6x5

- 10x4+ 30x3

-5x5+ 15x4

+ 25x3- 75x2

-10x4+ 30x3

+ 50x2- 150x

= x7- x6

-16x5+ 10x4

+ 85x3- 25x2

- 150x;

domain: all real numbers 78. =

= =

= =

= ; domain: all real numbers except 3,

, and - 79. = =

= =

= = ; domain: all real

numbers except 0, -2, , and - 80. f( f(x)) =

f = = ? = x 81. f( f( f(x))) =

= f(x) = 82. f(f(f(x))) = =

= = =

= = =

= 83. =

= =

= 2 84. =

=

= 4 85. The total cost is p(x) + c(x) = 1.5x + 3 +

0.75x + 2 = 2.25x + 5; the answer is B. 86. f(x) - g(x)= 2(x - 2)2

- (x2- 5x + 4 = 2(x2

- 4x + 4) - (x2-

5x + 4 = 2x2- 8x + 8 - x2

+ 5x - 4 = x2- 3x + 4;

the answer is D. 87. At 2 hours, the handymen chargethe same amount. After 2 hours, Tim’s cost is less thanAndy’s cost. The answer is D. 88. =

x + 1; x2+ 3 = (x + 1)2; x2

+ 3 = x2+ 2x + 1; 2x =

2; x = 1 89. x + 8 = ; (x + 8)2= x2

+

16; x2+ 16x + 64 = x2

+ 16; 16x = -48; x = -390. = x + 1; x2

+ 9 = (x + 1)2; x2+ 9 =

x2+ 2x + 1; 2x = 8; x = 4 91. - x = -3;(x2 2 9)

12

"x2 1 9

(x2 1 16)12

"x2 1 3

4hh

4a 1 4h 2 1 2 4a 1 1h

4(a 1 h) 2 1 2 (4a 2 1)h 5

f(a 1 h) 2 f(a)h

2hh

2 1 2h 2 3 2 2 1 3h

2(1 1 h) 2 3 2 (2(1) 2 3)h

f(1 1 h) 2 f(1)h

6 2 x8

8 2 2 2 x8

2 2 2 1 x4

21 22 1 x

42fQ2 1 x

4 R

fa1 1 x2

2bfa2 2 1 1 x

22

bfa1 21 2 x

22

b

fQfQ1 2 x2RR1

xfQfQ1xRR

x1

11

11x

Q1xR!5!5

x 2 3x2 1 2x

x 2 3x(x 1 2)

(x2 2 5)(x 2 3)

x(x2 2 5)(x 1 2)

x2(x 2 3) 2 5(x 2 3)

x(x2(x 1 2) 2 5(x 1 2))x3 2 3x2 2 5x 1 15

x(x3 1 2x2 2 5x 2 10)

x3 2 3x2 2 5x 1 15x4 1 2x3 2 5x2 2 10x

g(x)f(x)!5!5

x2 1 2xx 2 3

x(x 1 2)x 2 3

x(x2 2 5)(x 1 2)

(x2 2 5)(x 2 3)

x(x2(x 1 2) 2 5(x 1 2))

x2(x 2 3) 2 5(x 2 3)

x(x3 1 2x2 2 5x 2 10)

x3 2 3x2 2 5x 1 15x4 1 2x3 2 5x2 2 10xx3 2 3x2 2 5x 1 15

f(x)g(x)

(x4 1 2x3 2 5x2 2 10x)


= x - 3; x2- 9 = (x - 3)2; x2

- 9 = x2-

6x + 9; 6x = 18; x = 3 92. - 2 = x;

= x + 2; x2+ 12 = (x + 2)2; x2

+ 12 =

x2+ 4x + 4; 4x = 8; x = 2 93. ;

3x = x + 6; 2x = 6; x = 3 94. (x + 4)8= x8

+

8x7(4) + 28x6(4)2+ 56x5(4)3

+ 70x4(4)4+ 56x3(4)5

+

28x2(4)6+ 8x(4)7

+ (4)8= x8

+ 32x7+ 448x6

+

3584x5+ 17,920x4

+ 57,344x3+ 114,688x2

+ 131,072x +

65,536 95. (x + y)6= x6

+ 6x5y + 15x4y2+ 20x3y3

+

15x2y4+ 6xy5

+ y6 96. (2x - y)4= (2x)4

+

4(2x)3(-y) + 6(2x)2(-y)2+ 4(2x)(-y)3

+ (-y)4=

16x4- 32x3y + 24x2y2

- 8xy3+ y4

97. (2x - 3y)7= (2x)7

+ 7(2x)6(-3y) +21(2x)5(-3y)2

+ 35(2x)4(-3y)3+ 35(2x)3(-3y)4

+

21(2x)2(-3y)5+ 7(2x)(-3y)6

+ (-3y)7= 128x7

-

1344x6y + 6048x5y2- 15,120x4y3

+ 22,680x3y4-

20,412x2y5+ 10,206xy6

- 2187y7

98. (9 - 2x)5= 95

+ 5(9)4(-2x) + 10(9)3(-2x)2+

10(9)2(-2x)3+ 5(9)(-2x)4

+ (-2x)5= 59,049 -

65,610x + 29,160x2- 6480x3

+ 720x4- 32x5

99. (4x - y)5= (4x)5

+ 5(4x)4(-y) + 10(4x)3(-y)2+

10(4x)2(-y)3+ 5(4x)(-y)4

+ (-y)5= 1024x5

-

1280x4y + 640x3y2- 160x2y3

+ 20xy4- y5

100. (x2+ x)4

= (x2)4+ 4(x2)3(x) + 6(x2)2(x)2

+

4(x2)(x)3+ (x)4

= x8+ 4x7

+ 6x6+ 4x5

+ x4

101. (x2+ 2y3)6

= (x2)6+ 6(x2)5(2y3) +

15(x2)4(2y3)2+ 20(x2)3(2y3)3

+ 15(x2)2(2y3)4+

6(x2)(2y3)5+ (2y3)6

= x12+ 12x10y3

+ 60x8y6+

160x6y9+ 240x4y12

+ 192x2y15+ 64y18

102. = (2 - 3(2i)) +(4 + 2(4i)) = 2 - 6i + 4 + 8i = 6 + 2i103. =

= = -2 +

104. - = -

= =

105. = =

10 - 30i - 6i + 18i2= 10 - 18 - 30i - 6i = -8 - 36i

CHECKPOINT QUIZ 2 page 410

1. = = (-3x)4=

81x4 2. = = =

= 3. -

4 = 0; = 4; 3x + 1 = 42; 3x + 1 = 16; 3x =

15; x = 5 4. ; ; 5x + 2 =

27; x = 5 or x = - 5. - 3 = 3x;= 3x + 3; 3x + 3 = (3x + 3)2; 3x + 3 =

9x2+ 18x + 9; 9x2

+ 15x + 6 = 0; 3x2+ 5x + 2 = 0;

(3x + 2)(x + 1) = 0; 3x + 2 = 0 or x + 1 = 0; x =

or x = -1 6. (2 - x)0.5- x = 4; (2 - x)0.5

=223

!3x 1 3!3x 1 329

54

5x 1 2 5 932(5x 1 2)

23 5 9

!3x 1 1

!3x 1 11(2y)2 5

14y2

1("5 32y5)2 5

1("5 (2y)5)2

1(32y5)

25

1(32y5)0.4(32y5)20.4

("3 (23x)3 )

45 ("3 227x3

)4

(227x3)43

(5 2 3i)(2 2 6i)(5 2 !29)(2 2 !236)13 1 5i!56 1 2i!5 1 7 1 3i!5(27 2 3i!5)

(6 1 2i!5)(27 2 !245)(6 1 !220)19i!215i!2 2 2 1 4i!2(2 2 4i!2)

3(5i!2) 23!250 2 (2 2 !232)

(2 2 3!24) 1 (4 1 2!216)

(3x)12 5 (x 1 6)

12

"x2 1 12

"x2 1 12

"x2 2 9 x + 4; 2 - x = (x + 4)2; 2 - x = x2+ 8x + 16; x2

+

9x + 14 = 0; (x + 7)(x + 2) = 0; x + 7 = 0 or x + 2 =

0; x = -7 (extraneous) or x = -2 7.

+ 3 + = 1 + 3 + =

8. (f ? g)(1) = f(1) ? g(1) = (2(1) + 3) =

5(0) = 0 9.

10. ( f + g)(5) = f(g(5)) = = f(20) = 2(20)

+ 3 = 43

GUIDED PROBLEM SOLVING page 411

1. (C + r)(x) when x = 3: C(r(3)) = C(12.5(3)) =C(37.5) = 2p(37.5) = 75p; 3 seconds after it isformed a ripple will have a circumference of 75p in.,or about 236 in. 2. (A + d)(x) when x = 10.5:A(d(10.5)) = A(2.3(10.5)) = A(24.15) = p(24.15)2

< 687.09; 687.09 km2

7-7 Inverse Relations and Functions pages 412– 418

Check Skills You’ll Need For complete solutions seeDaily Skills Check and Lesson Quiz Transparencies orPresentation CD-ROM.1. 2.

3. 4.

5. 6.

�2 2 4Ox

y

2

�2

�4

x

y

2 4 6O

4

2

�2

�2 2O x

y

2

�22 4O

x

y

4

2

�6 2 8O x

y

8

2�4 4O x

y

4

�4

38

f(52 2 5)

2(2) 1 3

22 2 25 7

2QfgR(2) 5f(2)g(2) 5

(12 2 1)

154

14 2

12QQ12R

22 1

2R2Q12R(f 1 g)Q12R 5


Activity p. 412 1a. f(10) = 2(10) - 8 = 12;

g( f(10)) = g(12) = 1b. f(0) = 2(0) -

8 = -8; g(f(0)) = g(-8) = 1c. f(-7) =

2(-7) - 8 = -22; g( f(-7)) =

1d. g( f(-1496)) = -1496 2a. g(6) = ;

f(g(6)) = f(7) = 2(7) - 8 = 6 2b. g(0) = ;

f(g(0)) = f(4) = 2(4) - 8 = 0 2c. g(-32) =

; f(g(-32)) = f(-12) = 2(-12) - 8 =

-32 2d. f(g(p)) = p 3a. x = 2y - 8; 2y = x + 8;

3b. y = 2x - 8 and y =

They are reflections of each other in the line y = x.

CA Standards Check pp. 413–415 1a. The line y =

x is the perpendicular bisector of each segmentconnecting a point in s to the corresponding point in theinverse of s. The graph of the inverse of s is a reflectionin the line y = x of the graph of s. 1b. yes; no2a. Yes; no; for every x-value except 3 in the domain ofthe inverse there are two y-values. 2b. x = 3y - 10;

3y = x + 10; ; yes, it is a function becausefor each value of x there is only one y-value.3. y = 3x - 10; inverse: x = 3y - 10; 3y = x + 10;

4a. domain: all real numbers; range: all real numbers4b. f(x) = 10 - 3x, or y = 10 - 3x; inverse: x = 10 -

3y, 3y = -x + 10, y = , f -1(x) =

4c. domain: all real numbers; range: all real numbers

4d. f-1( f (3)) = f-1(10 - 3(3)) = f-1(1) =

4e. = =

5. ; 64d = v2; v =

8 � 44; about 44 ft/s!d 5 8!30

d 5 v2

64fQ83R 5 10 2 3Q83R 5 2

fQ22 1 103 Rf(f21(2))21 1 10

3 5 3

2x 1 103

2x 1 103

y 5 13x 1 10

3

y 5 13x 1 10

3

12x 1 4

y 5 x 1 82

232 1 82 5 212

0 1 82 5 4

6 1 82 5 7

222 1 82 5 27

28 1 82 5 0

12 1 82 5 10

6. + (777) = 777; + (-5802) = -5802

Exercises pp. 416–418

1.

2.

3.

4.

5. y = 3x + 1; inverse: x = 3y + 1; 3y = x - 1;; yes 6. y = 2x - 1; inverse: x = 2y - 1;

2y = x + 1; ; yes 7. y = 4 - 3x; inverse:

x = 4 - 3y; 3y = -x + 4; ; yes 8. y =

5 - 2x2; inverse: x = 5 - 2y2; 2y2= 5 - x; ;

y = no 9. y = x2+ 4; inverse: x = y2

+ 4;

y2= x - 4; ; no 10. y = 3x2

- 5;

inverse: x = 3y2- 5; 3y2

= x + 5; y2=

no 11. y = (x + 1)2; inverse: x =

(y + 1)2; ; ; no12. y = (3x - 4)2; inverse: x = (3y - 4)2; =

3y - 4; ; ; no

13. y = (1 - 2x)2+ 5; inverse: x = (1 - 2y)2

+ 5;x - 5 = (1 - 2y)2; = 1 - 2y;

2y = ; ; no

14. y = 2x - 3; inverse:x = 2y - 3; 2y = x + 3;

x 5 12x 1 3

2

y 5 1 4 !x 2 521 4 !x 2 5

4!x 2 5

y 5 4!x 1 433y 5 4!x 1 4

4!xy 5 4!x 2 14!x 5 y 1 1

y 5 4Åx 1 5

3 ;

x 1 53 ;

y 5 4!x 2 4

4Å5 2 x

2 ;

y2 5 5 2 x2

y 5 2 13x 1 4

3

y 5 12x 1 1

2

y 5 13x 2 1

3

�2 Ox

y2

�2

✘✘✘✘

yx 2 2

022

�3 �2 �1

Ox✘

✘✘

✘

y8

4

4 8

yx

0 1 20 4

391

Ox✘

✘✘

✘y

4

2

2 4

yx

1 2 30 2

431

Ox✘

✘✘

✘y

4

2

2 4

yx

1 2 30 0

421

f21)(ff)(f21


�2 2 4O x

y

4

2

�2

�4 2 6�8 Ox

y

6

2

�4

�8

�2 2O x

y

2

�2

15. y = 3 - 7x; inverse:x = 3 - 7y; 7y = -x + 3;

16. y = -x; inverse:x = -y; y = -x

17. y = 3x2; inverse:x = 3y2; ; y =

18. y = -x2; inverse:x = -y2; y2

= -x;y =

19. y = 4x2- 2; inverse:

x = 4y2- 2; 4y2

= x + 2;

y2= ;

y =

20. y = (x - 1)2;inverse: x = (y - 1)2;

= y - 1;

21. y = (2 - x)2;inverse: x = (2 - y)2;

= 2 - y;y 5 2 4 !x4!x

y 5 1 4 !x4!x

4Åx 1 2

4 5 4!x 1 2

2

x 1 24

4!2x

4Åx3y2 5 x

3

y 5 217x 1 3

7

22. y = (3 - 2x)2- 1;

inverse: x = (3 - 2y)2-

1; x + 1 = (3 - 2y)2;

;

23. f (x) = 3x + 4, or y = 3x + 4; inverse: x = 3y + 4;

3y = x - 4; ; ; domain

and range for both f and f-1 are all real numbers; f-1 isa function. 24. f (x) = , or y = ;inverse: x = ; x2

= y - 5; y = x2+ 5;

f-1(x) = x2+ 5, x � 0; domain of f {xΔ x � 5}, range

of f {yΔy � 0}; domain of f-1 {xΔx � 0}, range of f-1 {yΔy � 5}; f-1 is a function. 25. ,or ; inverse: ; x2

= y + 7; y =

x2- 7; f-1(x) = x2

- 7, x � 0; domain of f {xΔx � -7},range of f { yΔy � 0}; domain of f-1 {xΔx � 0},range of f-1 {yΔy � -7}; f-1 is a function.26. , or ; inverse:

; x2= -2y + 3; 2y = 3 - x2;

y = , x � 0; domain of f

xΔx � , range of f {yΔy � 0}; domain of f-1

{xΔx � 0}, range of f-1 yΔy � ; f-1 is a function.

27. f(x) = 2x2+ 2, or y = 2x2

+ 2; inverse: x = 2y2+

2; 2y2= x - 2; y2

= ; y = f-1(x) =

, x � 2; domain of f { all real numbers}, range

of f { yΔy � 2}; domain of f-1 {xΔx � 2}, range of f-1

{all real numbers}; f-1 is not a function. 28. f(x) =-x2

+ 1, or y = -x2+ 1; inverse: x = -y2

+ 1;y2

= 1 - x; ; f-1(x) = , x � 1;domain of f {all real numbers}, range of f {yΔy � 1};domain of f-1 {xΔx � 1}, range of f-1 {all real numbers};f-1 is not a function.

29a. ; solve for ;

; yes 29b.

30a. ; solve for r: ;

r = yes 30b. r = � 20.29; 20.29 ft

31. (f-1 + f )(10) = 10 32. (f + f-1)(-10) = -1033. (f-1 + f )(0.2) = 0.2 34. ( f + f-1)(d) = d35. f (x) = 1.5x2

- 4, or y = 1.5x2- 4;

inverse: x = 1.5y2- 4; 1.5y2

= x + 4; ;

no

36. , or ; inverse: ; 4x = 3y2;

; f-1(x) = no42Åx3;y 5 4Å

43x 5 42Å

x3;y2 5 4

3x

x 53y2

4y 5 3x2

4f(x) 5 3x2

4

f21(x) 5 4Å2x 1 8

3 ;y 5 4Å2x 1 8

3 ;

y2 52(x 1 4)

3

Å3 3(35,000)

4pÅ3 3V

4p ;

r3 5 3V4pV 5 4

3pr323.898F

F 5 59(25 2 32) < 23.89;F 5 5

9(C 2 32)

F: 95F 5 C 2 32C 5 95F 1 32

4!1 2 xy 5 4!1 2 x

4Åx 2 2

2

4Åx 2 2

2 ;x 2 22

32VU

32VU

3 2 x2

2 ; f21(x) 5 3 2 x2

2

x 5 !22y 1 3y 5 !22x 1 3f(x) 5 !22x 1 3

x 5 !y 1 7y 5 !x 1 7f(x) 5 !x 1 7

!y 2 5!x 2 5!x 2 5

f21(x) 5 x 2 43y 5 x 2 4

3

y 5 3 4 !x 1 12

2y 5 3 4!x 1 14!x 1 1 5 3 2 2y;


2 4Ox

y4

2

�2

�2 2O x

y

2

�2

x

y

4O

4

2

�2

�2�4�6O x

y2

�2

�4

2 4�2O x

y4

2

�2

8Ox

y8

4

4Ox

y

4

4 6Ox

y

6

4

37. f (x) = , or ; inverse:

; ; (x - 3)2=

2y - 1; 2y = x2- 6x + 9 + 1; ;

, x � 3; yes 38. f (x) = (x + 1)2,

or y = (x + 1)2; inverse: x = (y + 1)2; ;

; f-1(x) = ; no39. f (x) = (2x - 1)2, or y = (2x - 1)2; inverse: x =

(2y - 1)2; 2y - 1; ; ;

; no 40. f (x) = (x + 1)2- 1, or

y = (x + 1)2- 1; inverse: x = (y + 1)2

- 1; x + 1 =

(y + 1)2; = y + 1; y = -1 ;f-1(x) = -1 ; no 41. f (x) = x3, or y = x3;inverse: x = y3; y = ; f-1(x) = ; yes

42. f (x) = x4, or y = x4; inverse: x = y4; = y;

f-1(x) = ; no 43. , or

, inverse: ; ;

5(x - 1) = 2y2; y2= ;

no 44. ; v2= 2gx;

; flow is 40 ft/s: x = ;

flow is 20 ft/s: x = = 6.25; 6.25 ft 45. The range ofthe inverse is the domain of f, which is the set of all realnumbers x � 1. 46. 2 and 5 47. f(x) = , or

; inverse: ; ; y = x2;

f-1(x) = x2, x � 0; domain of f {xΔx � 0}, range of f{yΔy � 0}; domain of f-1 {xΔx � 0}, range of f-1

{yΔy � 0}; f-1 is a function. 48. f (x) = , or

; inverse: ; ;

y = (x - 3)2; f-1(x) = (x - 3)2, x � 3; domain of f{xΔx � 0}, range of f {yΔy � 3}; domain of f-1

{xΔx � 3}, range of f-1 {yΔy � 0}; f-1 is a function.

49. f (x) = , or ; inverse:

; x2= -y + 3; y = 3 - x2; f-1(x) =

3 - x2, x � 0; domain of f {xΔx � 3}, range of f{yΔ y � 0}; domain of f-1 {xΔx � 0}, range of f-1

{yΔy � 3}; f-1 is a function. 50. f (x) = ,

or ; inverse: ; x2= y + 2;

y = x2- 2; f-1(x) = x2

- 2, x � 0; domain of f

{xΔx � -2}, range of f {yΔy � 0}; domain of f-1

{xΔx � 0}, range of f-1 {yΔy � -2}; f -1 is a function.

51. f (x) = , or ; inverse: ; 2x = y2;

; f-1(x) = , x � 0; domain of f{all real numbers}, range of f {yΔy � 0}; domain of f -1

{xΔx � 0}, range of f-1 {all real numbers}; f-1 is not a

function. 52. f (x) = , or ; inverse: ;

xy2= 1; ; , x � 0; domain f21(x) 5 1

!xy 5 Å

1x;y2 5 1

x

x 5 1y2y 5 1

x21x2

4!2xy 5 4!2x

x 5y2

2y 5 x2

2x2

2

x 5 !y 1 2y 5 !x 1 2

!x 1 2

x 5 !2y 1 3

y 5 !2x 1 3!2x 1 3

x 2 3 5 !yx 5 !y 1 3y 5 !x 1 3

!x 1 3

2x 5 !yx 5 2!yy 5 2!x2!x

202

64

402

64 5 25; 25ftx 5 v2

2g 5v2

2(32) 5v2

64

v 5 !2gxf21(x) 5 4Å5x 2 5

2 ;

y 5 4Å5x 2 5

2 ;5x 2 52

x 2 1 52y2

5x 52y2

5 1 1y 5 2x2

5 1 1

f(x) 5 2x2

5 1 14!4 x

4!4 x

!3 x!3 x4 !x 1 1

4 !x 1 14!x 1 1

f21(x) 5 1 4 !x2

y 5 1 4 !x22y 5 1 4 !x4!x 5

4!x 2 1y 5 4!x 2 1

4!x 5 y 1 1

f21(x) 5 x2 2 6x 1 102

y 5 x2 2 6x 1 102

x 2 3 5 !2y 2 1x 5 !2y 2 1 1 3

y 5 !2x 2 1 1 3!2x 2 1 1 3 of f {xΔx 2 0}, range of f {yΔy � 0}; domain of f-1

{xΔx � 0}, range of f-1 {yΔy 2 0}; f-1 is not a function.53. f (x) = (x - 4)2, or y = (x - 4)2; inverse: x =

(y - 4)2; ; ; f-1(x) =; domain of f {all real numbers}, range of f

{yΔy � 0}; domain of f-1 {xΔx � 0}, range of f-1

{all real numbers}; f-1 is not a function. 54. f (x) =(7 - x)2, or y = (7 - x)2; inverse: x = (7 - y)2;

; ; f-1(x) = , x � 0;domain of f {all real numbers}, range of f {yΔy � 0};domain of f-1 {xΔx � 0}, range of f-1 {all real numbers};

f-1 is not a function. 55. f (x) = , or

; inverse: ; x(y + 1)2= 1;

(y + 1)2= y + 1 =

, x � 0; domain of f {xΔx 2 -1},

range of f {yΔy � 0}; domain of f-1 {xΔx � 0}, range of f-1 { yΔy 2 -1}; f-1 is not a function. 56. f (x) =

, or y = ; inverse: x = ;

x - 4 = ; ; y = ;

f-1(x) = , x � 4; domain of f {xΔx � 0},

range of f {yΔy � 4}; domain of f-1 {xΔx � 4}, range

of f-1 {yΔy � 0}; f-1 is a function. 57. f (x) = , or

y = ; inverse: x = ; = 3; ; 2;

f-1(x) = , x � 0; domain of f {xΔx � 0}, range of f

{yΔy � 0}; domain of f-1 {xΔx � 0}, range of f-1

{yΔy � 0}; f-1 is a function. 58. f (x) = , or

inverse: x = ;

; -2y ; ; f-1(x) = ,

x � 0; domain of f {xΔx � 0}, range of f {yΔy � 0};domain of f-1 {xΔx � 0}, range of f-1 {yΔy � 0};f-1 is a function.

59a. Answers may vary. Sample:

59b. Answers may vary. Sample:

60. r is not a function because there are two y-valuesfor one x-value. r-1 is a function because each of its x-values has one y-value. 61. s2

+ s2= h2; h2

= 2s2;; ; inverse (solve for s): ;s!2 5 hh 5 s!2h 5 "2s2

212Q

1xR

2y 5 21

2Q1xR

25 Q1xR

2!22y 5 1x

x!22y 5 1;1!22y

y 5 1!22x

;

1!22x

Q3xR2

y 5 Q3xR!y 5 3xx!y3

!y3!x

3!x

Q2 x 2 42 R2

Q2 x 2 42 R2!y 5 2x 2 4

222!y

4 2 2!y4 2 2!x4 2 2!x

f21(x) 5 4Å1x 2 1

4Å1x; y 5 4Å

1x 2 1;1

x;

x 5 1(y 1 1)2y 5 1

(x 1 1)2

1(x 1 1)2

7 4 !xy 5 7 4 !x4!x 5 7 2 y

4!x 1 4y 5 4!x 1 44!x 5 y 2 4


�2024

04

16

059

�4�2

04

; ; hypotenuse of 6 in.:

� 4.2; about 4.2 in. 62. Check students’ work.

63. , or ; inverse: ; y3= 5x;

; f -1(x) = ; yes 64. f (x) = , or

y = ; inverse: x = ; x3= y - 5; y =

x3+ 5; f-1(x) = x3

+ 5; yes 65. f(x) = , or y =

; inverse: x = ; 3x = ; (3x)3= y; f-1(x) =

27x3; yes 66. f(x) = (x - 2)3, or y = (x - 2)3;

inverse: x = (y - 2)3; = y - 2; y = 2 + ;

; yes 67. , or ;

inverse: ; x4= y; f-1(x) = x4, x � 0; yes

68. f(x) = 1.2x4, or y = 1.2x4; inverse: x = 1.2y4;

; y = ; no

69. (f ° g)(x) + h(x) = 4 x + 7 + |-2x + 4| =

2x + 28 + |-2x + 4|; the answer is C. 70. Since f-1

and f are inverses, (f -1° f)(x) = x for all values of x;(f -1° f)(10) = 10; the answer is B. 71. Use Pascal’sTriangle to expand (a + b)4

= a4+ 4a3b + 6a2b2

+

4ab3+ b4; substitute x3 for a and x for b; (x3)4

+

4(x3)3(x) + 6(x3)2(x)2+ 4(x3)(x)3

+ (x)4= x12

+ 4x10

+ 6x8+ 4x6

+ x4; the answer is A.

72. ( f ° g)(x) = f(g(x)) = f + 7 = 4 + 7 =

2x + 28 73. (g ° f )(x) = g( f (x)) = g(4x) = (4x) +

7 = 2x + 7 74. (h ° g)(x) = h(g(x)) = h + 7 =

= Δ-x - 14 + 4Δ = Δ-x - 10Δ

75. g(x) + g(x) = + 7 + + 7 = x + 14

76. (h ° (g ° f))(x) = h(g( f (x))) = h(g(4x)) =

h + 7 = h(2x + 7) = Δ-2(2x + 7) + 4Δ =

Δ-4x - 14 + 4Δ = Δ-4x - 10Δ 77. ( f ° g)(x) +

h(x) = f(g(x)) + h(x) = f x + 7 + h(x) =

4 + Δ-2x + 4Δ = 2x + 28 + Δ-2x + 4Δ

78. = = 2 79. - = - = -2

80. ; not a real number 81. = = 3

82. - = - = -3 83. =

= -3 84. = = 0.4

85. = = 30

86. 2x3+ 3x2

- 8x - 12 = 0; possible rational roots:

41, 42, 43, 44, 46, 412, 4 , 4 ;

Test 1: 2(1)3+ 3(1)2

- 8(1) - 12 = -15 2 0;Test 2: 2(2)3

+ 3(2)2- 8(2) - 12 = 0, so 2 is a root.

2 3 -8 -124 14 12

2 7 6 0

2;

32

12

"4 304!4 810,000

"3 (0.4)3!3 0.064"5 (23)5

!52243"5 35!5 243

"5 35!5 243!4 216

"4 24!4 16"4 24!4 16

Q12x 1 7RRQ12

RQ12(4x)

12x1

2x

P22Q12x 1 7R 1 4 PRQ12x

12

RQ12xRQ12x

R12Q

4Å4 5x

6y4 5 5x6

x 5 !4 y

y 5 !4 xf(x) 5 !4 xf21(x) 5 2 1 !3 x

!3 x!3 x

!3 y!y3

3!x3

3

!x3

3

!3 y 2 5!3 x 2 5

!3 x 2 5!3 5xy 5 !3 5x

x 5 15y3y 5 1

5x3f(x) 5 15x3

3!2

s 5 6!22 5s 5 h!2

2s 5 h!2

2x2+ 7x + 6 = 0; (2x + 3)(x + 2) = 0; 2x + 3 = 0

or x + 2 = 0; x = - or x = -2; roots are 2, -2, -

87. 3x3- 5x2

- 4x + 4 = 0; possible rational roots:41, 42, 44, 4 , 4 , 4 ;

Test 1: 3(1)3- 5(1)2

- 4(1) + 4 = -2;Test -1: 3(-1)3

- 5(-1)2- 4(-1) + 4 = 0, so -1

is a root.3 -5 -4 4

-3 8 -43 -8 4 0

3x2- 8x + 4 = 0; (3x - 2)(x - 2) = 0; 3x - 2 = 0 or

x - 2 = 0; x = or x = 2; roots are -1, 2,

88. 3x3+ 10x2

- x - 12 = 0; possible rational roots:41, 42, 43, 44, 46, 412, 4 , 4 , 4 ;

Test 1: 3(1)3+ 10(1)2

- 1 - 12 = 0, so 1 is a root.3 10 -1 -12

3 13 123 13 12 0

3x2+ 13x + 12 = 0; (3x + 4)(x + 3) = 0; 3x + 4 = 0

or x + 3 = 0; x = - or x = -3; roots are 1, -3, -

89. 2x3- 11x2

- x + 30 = 0; possible rational roots:

41,42,43,45,46,410,415,430,4 ,4 ,4 ,4 ;

Test 1: 2(1)3- 11(1)2

- 1 + 30 = 20 2 0;Test 2: 2(2)3

- 11(2)2- 2 + 30 = 0, so 2 is a root.

2 -11 -1 304 -14 -30

2 -7 -15 02x2

- 7x - 15 = 0; (2x + 3)(x - 5) = 0; 2x + 3 = 0 or

x - 5 = 0; x = - or x = 5; roots are 2, 5, -

90. x3- 6x2

+ 11x - 6 = 0; possible rational roots:41, 42, 43, 46;Test 1: 13

- 6(1)2+ 11(1) - 6 = 0, so 1 is a root.

1 -6 11 -61 -5 6

1 -5 6 0x2

- 5x + 6 = 0; (x - 3)(x - 2) = 0; x - 3 = 0 or x - 2 = 0; x = 3 or x = 2; roots are 1, 2, 391. x3

+ 3x2- 4x - 12 = 0; possible rational roots:

41, 42, 43, 44, 46, 412;Test 1: 13

+ 3(1)2- 4(1) - 12 = -12 2 0;

Test 2: 23+ 3(2)2

- 4(2) - 12 = 0, so 2 is a root.1 3 -4 -12

2 10 121 5 6 0

x2+ 5x + 6 = 0; (x + 3)(x + 2) = 0; x + 3 = 0 or

x + 2 = 0; x = -3 or x = -2; roots are 2, -2, -3

2;

1;

32

32

2;

152

52

32

12

43

43

1;

43

23

13

23

23

21;

43

23

13

32

32


ACTIVITY LAB page 419

1. y = x2- 5

2. y = (x - 3)2

3. y = 0.01x4

4. y = 0.5x3- 3

5. The third graph would be the same as the first.Interchanging the pairs twice restores the original pairs.6. The graph is the reflection of the first graph in the x-axis. 7. Answers may vary. Sample: Usingparametric equations, you only have to change Y1T tosee a new function and its inverse.

7-8 Graphing Square Root and OtherRadical Functions pages 420–425

Check Skills You’ll Need For complete solutions seeDaily Skills Check and Lesson Quiz Transparencies orPresentation CD-ROM.

1. 2.

642Ox

y

4

2

�2�4�6 Ox

y

4

3. 4.

5. 6.

CA Standards Check pp. 421–423

1. y = - 3 and y =

2. y = and y =

3. y = - 4

4. y = 3 - !3 x 1 1

14!x 2 2

!x 1 4!x 2 1

!x 1 3!x

�2 2Ox

y

2

�4 Ox

y

�2

�4

2

�2 2Oxy

�2

�4

�2�4�6 O xy

�2

�4

�6


1 3�3 Ox

y3

1

�1

�3

84Ox

y8

4

2�2 Ox

y

2

�2

2�2�4 Ox

y

�2

�4

�6

642O xy

�2

�4

�6

2 4 6�2�4 Ox

y5

3

1

2 4Ox

y4

2

�2

�1 1 3�3O x

y3

1

5. Graph Y1 = and Y2 = 3.5. Use theIntersect feature to find the x-coordinate of theintersection.

The height for 3.5-seconds isabout 60 m. This is about theheight of the 7-second fall.

6. y = + 3 = + 3 =

+ 3; the graph of y = + 3 is the

graph of y = translated 3 units right and 3 units up.

Exercises pp. 423–425

1. y = + 1 2. y = - 2

3. y = - 4 4. y = + 5

5. y = 6. y =

7. y = 8. y =

9. y = 10. y = -0.25!x3!x

!x 2 4!x 1 6

!x 1 1!x 2 3

!x!x

!x!x

2!3 x

2!3 x 2 32!3 x 2 3

!3 8(x 2 3)!3 8x 2 24

14

IntersectionX�60.08625 Y�3.5

!(2X>9.81) 11. y = 12. y =

13. y = -5 + 2 14. y = -0.75 + 3

15. y = + 2

16. y = - 1 17. y = 3 + 4

18. y =

19. y = !3 x 2 4

!3 x 1 5

!x 1 114!x 1 2

2!x 2 3

642O x

y

2

44 8

O xy

!x!x 2 3

2!x 2 113!x


642Ox

y6

4

24O x

y4

2

�2

�4

2 4O x

y

642Ox

y

6

4

2

642Ox

y

2

642Ox

y

2

�2�4�6 Ox

y3

1

6 842Ox

y

2

642Ox

y6

4

2

21O x

y

�1

1Ox

y

1

642Oxy

�2

�4

1284Ox

y

12

8

4

642Ox

y

�2

2

642Oxy

�1

2�2�4 Ox

y

�2

1

2 6Ox

y2

�2

20. y = - 7 21. y = - 1

22. y = - 9 23. y = + 3

24a. y = 117.75

24b. Trace the graph to x = 40, 80, and 130; �745 ft,�1053 ft, �1343 ft

25. The intersection is at x = 147, so the solutionis 147.

26. The intersection is at x = 9.5, so the solutionis 9.5.

27. The intersection is at x = -8.11, so thesolution is -8.11.

28. no solution; The left-hand side is never negative andthe right-hand side is always negative.

IntersectionX�-8.111111 Y�10

IntersectionX�7.5 Y�4

IntersectionX�147 Y�12

!x

12!

3 x 2 12!3 x 2 6

2!3 x 1 3!3 x 1 2 29. The intersection is at x = 5, so the solution is 5.

30. The intersection is at x = -1, so the solutionis -1.

31. y = = = ; the graph of y = is the graph of y =

translated 1 unit to the right. 32. y = - =

- = -4 ; the graph of y =

is the graph of y = -4 translated 2 units to the left. 33. y = -2 =

-2 = ; the graph of y =

-14 is the graph of y = -14 translated 1

unit to the left. 34. y = = =

; the graph of y = is the graph of

translated 2 units to the left.

35. y = - 3 = - 3 =

- 3; the graph of y = - 3 is the

graph of y = translated 2 units to the right and

3 units down. 36. y = + 1 =

+ 1 = + 1; the graph of

y = + 1 is the graph of y = translated 2units to the right and l unit up.

37. y = + 7

domain: x � 0; range: y � 7

38. y = - 6

domain: x � 0;range: y � -6

39. y =

domain: x � 6;range: y � 0

!x 2 6

!x

!x

3!3 x3!3 x 2 2

3!3 x 2 2!3 27(x 2 2)

!3 27x 2 54

8!x

8!x 2 28!x 2 2

!64(x 2 2)!64x 2 128

4!3 x

4!3 x 1 24!3 x 1 2

!3 64(x 1 2)!3 64x 1 128

!x!x 1 1214!x 1 1!49(x 1 1)

!49x 1 49!x24!x 1 2

!x 1 2!16(x 1 2)!16x 1 32

3!x3!x 2 13!x 2 1!9(x 2 1)!9x 2 9

IntersectionX -1 Y 1.7320508

IntersectionX 5 Y 6


1208040 160OA

1200

800

400

l

O xy

2�2�6 Ox

y

�2

�4

2

2 4 6 8Oxy

�4

�8�2 2O x

y4

2

1284 16Ox

y4

2

1284Ox

y12

8

4

642 8Oxy

�2

�4

�6

40. y = -3 + 2


41. y =


42. y = 7 -

domain: x � ;

range: y � 7

43. y = + 1

domain: all real numbers;range: all real numbers

44. y = + 3


45. y = - 3


23!3 x 2 4

12!x 2 1

4!3 x 2 2

12

!2x 2 1

245!x

!x 46. y = -

domain: x � ;

range: y � 0

47. y = + 5


48. y =

domain: all real numbers; range:all real numbers

49. y =

domain: x � -5;range: y � -1

50. y =


51. y = + 7

domain: x � ;

range: y � 7

34

23Åx 2 34

4 2 !3 x 1 2.5

21 2 !4x 1 20

22!3 x 2 4

2!3 8x

212

Åx 1 12


642 8Ox

y

�2

�4

�6

2

642 8Oxy

�1

�2

642 8Ox

y

6

4

2

4Ox

y

4

2

�2

642 8 10Ox

y4

2

128Ox

y

�4

�8

�4

4

1284 16Oxy

�2

�4

�2 2 4�4 Ox

y8

4

2

62 8Ox

y

�2

�4

4

2

�2 2 4�4 Oxy

�2

�4

�6

�2�4 Ox

y

4

2

642Ox

y6

4

2

52a. y = 1.11

52b. t = 1.11 � 4.3; about 4.3 s;t = 1.11 � 6.1; about 6.1 s53a. y = - 2

53b. domain: x � 2; range: y � -2 53c. (2, -2)53d. The domain is based on the x-coordinate of thatpoint, and the range is based on the y-coordinate.54a. y = - 2 54b. y = -

2 - 3; y = - 5

55a. y = + 1 and y = - + 1

55b. Both domains are x � 2. The range of y =

+ 1 is y � 1. The range of y = - + 1

is y � 1. 56. y = - 1; y =

- 1; y = 5 - 1; the graph is the

same as y = 5 , translated 4 units right and 1 unit

down. 57. y = + 4; y = 36 +

4; y = + 4; the graph is the graph of y =

, translated 3 units left and 4 units up. 58. y =

- ; y = ;

the graph is the graph of y = - , translated unit

right. 59. y = - 2; y = - 2;

y = - 2; the graph is the same as y = ,

translated 1 unit right and 2 units down. 60. y = 10 -

; y = 10 - ; y = 10 - ;

the graph is the same as y = - , translated 3 units

left and 10 units up. 61. y = + 5; y =

+ 5; y = + 5; the graph is the

same as y = , translated 9 units left and 5 units up.

62. Answers may vary. Sample: y = !3 x 2 2 1 4

"x13

13!x 1 9Å

19(x 1 9)

Åx9 1 1

13!

3 x

13!

3 x 1 3Å3 1

27(x 1 3)Å3 x 1 3

27

12!x1

2!x 2 1

Å14 (x 2 1)Å

x 2 14

142!3 x

y 5 22"3 x 2 14;2Å

3 8Qx 2 14R!3 8x 2 2

6!x

6!x 1 3

!x 1 3!36x 1 108

!x

!x 2 4!25(x 2 4)

!25x 2 100

!x 2 2!x 2 2

!x 2 2!x 2 2

!x 2 1!x 2 5 1 4!x 2 5

!x 2 2!30

!15

!x 63a. y =

63b. d = = = 20; 20 in. 64. If a � 0,the graph is stretched vertically by a factor of a. If a � 0, the graph is reflected over the x-axis andstretched vertically by a factor of ΔaΔ. 65. y =

- ; y = - ; y = - ; the graph is the graph of y = - translated 4 units tothe left; domain: x � -4, range: y � 0 66. y =

- ; y = - ; y = - ;

y = - ; the graph is the graph of y = -

translated units to the right; domain: x � , range: y � 0

67. y = + 6; y = + 6;

y = ? + 6; the graph is the graph of

y = , translated units right and 6 units up;

domain: x � , range: y � 6 68. y = -3 - ;

y = -3 - ; y = - ? - 3;

the graph is the graph of y = - , translated units

left and 3 units down; domain: x � - , range: y � -3

69a.

69b. The graph of y = is a reflection of the

graph of y = in the line x = h. 70. for all

odd positive integers 71. Since the absolute value of

-3 is greater than , the graph of y = -3x2 is

narrower than y = x2. Also, the sign is negative so the

graph will open downward instead of upward. Theanswer is C. 72. The parent function of the graph is y = x2; the graph shows a horizontal translation of 1, avertical translation of 2 and a vertical stretch by a factorof ; the equation of the graph is y = (x - 1)2

+ 2; thevertex of the graph is (1, 2); the answer is B. 73.f(g(x)) = f(x + 3) = 2(x + 3)2

- 5 = 2(x2+ 6x + 9) -

5 = 2x2+ 12x + 18 - 5 = 2x2

+ 12x + 13; the answeris B. 74. f(x) = 4x - 1, or y = 4x - 1; inverse: x =

4y - 1; 4y = x + 1; ; yes 75. f(x) = ,

or y = ; inverse: x = ; = x + 3;

y = ; yes 76. f(x) = 2.4x2+ 1, or y =

2.4x2+ 1; inverse: x = 2.4y2

+ 1; 2.4y2= x - 1;

3(x 1 3)2

23y2

3y 2 323x 2 3

23x 2 3y 5 x 1 1

4

13

13

12

12

!x 2 h

!h 2 x

�4�6�8 �2 Ox

y4

2 2 � xy �

1 � xy �

�xy �

32

32!12x

Åx 1 32!12Å12Qx 1 3

2R!12x 1 185

3

53!3x

Åx 2 53!3

Å3Qx 2 53R!3x 2 5

34

34

!8xÅx 2 34!8

Å8Qx 2 34R!8x 2 6!2(4x 2 3)

!2x!x 1 4!2!2(x 1 4)!2x 1 8

!400!2(200)

!2x


642O t

y4

2

2 6O x

y

�2

8Ox

y

2

�2

642O A

d4

2

y2- ; y = ; no 77. f(x) = -

4, or y = - 4; inverse: x = - 4;

x + 4 = ; (x + 4)2= y + 3; y = (x + 4)2

-

3; yes 78. f (x) = (2x + 1)2, or y = (2x + 1)2;

inverse: x = (2y + 1)2; = 2y + 1;

2y = -1 ; y = ; no 79. f (x) = 2x3, or

y = 2x3, inverse: x = 2y3; y3= ; y = ; yes

80. = = 81. =

= 82. = ? =

83. = =

84. 5x2+ x = 3; 5x2

+ x - 3 = 0; x =

= 85. 3x2+

9x = 27; 3x2+ 9x - 27 = 0; x2

+ 3x - 9 = 0;

x = = =

86. x2- 9x + 15 = 0; x =

=

87. x2+ 10x + 11 = 0; x = =

= = 88. x2-

12x + 25 = 0; x = =

= = 89. 8x2+ 2x -

15 = 0; x = = =

; x = = or x = =

ACTIVITY LAB page 426-427

Activity 11. The domain of y = is x � 0. 2a. the differencebetween a number and its square root 2b. Y3 valuesincrease 3a. x-values are less than values. 3b.negative values 4a-c. Check students’ work.

Activity 25a. x � 3; x � 7 5b. 3 � x � 7 5c. 3 � x � 55d. Check students’ work. 6. yes, x � 3; When x � 3,

is an imaginary number.Activity 37. Check students’ work. 8. The screen shows ahorizontal segment at y = 1; 3 � x � 5; yes 9. 5 � x� 7; the graph does not show y = 1 for these x-values.10a-b. 1 � x � 5 10c. Check students’ work.11a. 3 � x � 5.4 11b. - � ;

x - 2 + x - 3 = 6 - x; -2 = 9"x2 2 3x"x2 2 3x

!6 2 x!x 2 3!x

!x 2 3

!x

!x

232

22 2 2216

54

22 1 2216

22 4 2216

22 4 !48416

22 4 "22 2 4(8)(215)2(8)

6 4 !1112 4 2!112

12 4 !442

2(212) 4 "(212)2 2 4(1)(25)2(1)

25 4 !14210 4 2!142

210 4 !562

210 4 "102 2 4(1)(11)2(1)

9 4 !212

2(29) 4 "(29)2 2 4(1)(15)2(1)

23 4 3!52

23 4 !452

23 4 "32 2 4(1)(29)2(1)

21 4 !6110

21 4 "12 2 4(5)(23)2(5)

"5 48x3y4

2yÅ5

3x3 ? (2y)4

2y ? (2y)4Å5 3x3

2y!3 9xy2

3y

"3 (3y)2

"3 (3y)2!3 x!3 3y

!3 x!3 3y

"3 12xy2

2yÅ3 3x ? (2y)2

2y ? (2y)2

Å3 3x

2y"3x2 5 x!3Å36x312x

"36x3

!12x

Å3 x

2x2

21 4 !x24 !x

4!x

!y 1 3

!y 1 3!x 1 3

!x 1 34Åx 2 1

2.4x 2 1

2.4- x; 4(x2

- 3x) = 81 - 54x + 9x2; 4x2- 12x = 9x2

- 54x + 81; 0 = 5x2- 42x + 81; x =

= =

= = = 5.4 or = 3; 3 � x � 5.4

11c. Answers may vary. Sample: it is difficult to find theendpoint of the true section of the graph. 12a. Theserestrictions ensure the radicands are nonnegative.12b. The graph will show part of y = 1 above thedomain of the inequality. 12c. The solution will be x-values for which Y2 is beneath Y3, within the intervalshown by Y1.

TEST-TAKING STRATEGIES page 428

1. I: = ( )x; = ( )2

is true; = ( )-2

is true; II: = ; = is true; =

is not true; III: = ΔxΔ+ 1; =

Δ2Δ+ 1 is not true; = Δ-2Δ+ 1 is not true;

only statement I is true for both values of x.

2. = ( )x; = ( )0

is true; = ( )1

is true; II: = ; = ; is true;

= is true; III: = ΔxΔ+ 1;

= Δ0Δ+ 1 is true; = Δ1Δ+ 1 is not true; since

III is not true when x = 1, choice B, D, and E could be

eliminated as correct answers.3. Check each equation with the value x = 1; I: =

x; = x is true; II: = x; = 1 is true; III:

= x; = 1 is true; all three expressions are

equal to x when x = 1; the correct answer would be E.

4. I: ( )8= = = 31

= 3; statement I

is true; II:

( )8= = = � 3; statement II is

not true; III: = � ; statement III is not true;

the answer is A.

CHAPTER REVIEW pages 429– 431

1. In the expression , 8 is called the radicand. 2. In

the expression , 3 is called the index. 3. When yourewrite an expression so there are no radicals in anydenominator and no denominators in any radical, yourationalize the denominator. 4. The expressions

and are not examples of like radicals. 5. The

definition of rational exponents allows us to write =

. 6. If g(x) = x - 4 and h(x) = x2, (g + h)(x) =x2

- 4 is a composite function. 7. To multiplyexpressions you sometimes add rational exponents.8. If f and f -1 are inverse functions, then (f ° f-1(x)) =

x and ( f-1 + f )(x) = x. 9. + 2 = x is an(x 2 7)12

"3 727

23

!5 x

!x

!3 8

!3 8

53

1Q53R1

1Q35R21

3343

12 ? 14 ? 16aQ31

2R14b

6

"3 "3

312 ? 14 ? 81aQ31

2R14b

8

"3 "3

"3 14

"3 1"3 x4

"3 x

"3 13"3 x3"12"x2

"12 1 1

"02 1 1"x2 1 1"3 13"12"3 03"02"3 x3"x2

"3 2"3 21"3 2"3 20"3 2"3 2x

"(22)2 1 1

"22 1 1"x2 1 1"3 (22)3"(22)2"3 23"22"3 x3"x2

"3 2"3 222"3 2"3 22"3 2"3 2x

3010

5410

42 4 1210

42 4 !14410

42 4 !1764 2 162010

2(242) 4 "(242)2 2 4(5)(81)2(5)


example of a radical equation. 10. The positive even

root of a number is called the principal root.

11. = = 12 12. =

= -0.4 13. = 7 14. =

= 0.5 15. = = -3

16. = = 7Δxy5Δ

17. = = -4y3 18. =

= (a - 1)2 19. = =

3x3 20. = = (y + 3)2

21. = =

4x4y2 22. ? = = =

= 20 23. ? = = =

= 24. ? =

= = 10x2y

25. ? = = =

7x2 26. ? = =

= 15x3y2 27. ?

= = =

3(2xy) = 6xy 28. = = =

= 4 29. = = =

= 2y 30. = = =

= 5x 31. = =

= = 6xy

32. = = =

= 3a2b 33. ? =

= = 34. ? = =

= 35. ? = =

36. = = ? =

37. = = ? =

38. + - = + 5 - 2 =

(3 + 5 - 2) = 6 39. (5 + )(2 - ) =10 - 5 + 2 - 3 = 7 - 340. (7 - )(7 + ) = 49 - 6 = 4341. - + = -

2 + 3 = 2

42. ? = =42 2 12!3

49 2 12(7 2 2!3)(7 2 2!3)

6(7 1 2!3)

!2x!2x!2x!2x!18x!8x!2x

!6!6!3!3!3

!3!3!3!3!3!33!3!12!75!27

y!3 150x10x2

!3 25x!325x

y!3 6

2x"3 5x2Å3 6y3

5x512

"3 6x2y4

2"3 5x7y

a3!ab4b

!b!b

a3!a4!bÅ

a7

16b"2a7b2

"32b3

"3 5x2

x2"3 5x2

"3 x6"3 x2

"3 x2!3 5"3 x4

x!6x4

x2!6x4x

"6x5

"16x2!2!2

"3x5

"8x22!3

34!3

6!48

6

!6!6

!8!6

!3 b"3 33 ? (a2)3 ? b3 ? b

"3 27a6b4

Å3 81a8b5

3a2b"3 81a8b5

"3 3a2b

!3x"62 ? 3 ? x2 ? x ? y2"108x3y2

Å216 x3y2

2"216x3y2

!2"(5x)2

"25x2Å

75x33x

"75x3

!3x!3 y"3 23 ? y3 ? y

"3 8y4Å3 56y5

7y"3 56y5

!3 7y"42

!16Å128

8!128!8

!4 2y!4 2y

3"4 24 ? 2 ? x4 ? y4 ? y3"4 32x4y5"4 8xy5

3"4 4x3!x"152(x3)2 ? x ? (y2)2

"225x7y4"45x3y"5x4y3!2

"72 ? 2 ? (x2)2"98x4!14x"7x3

"3 12y210"3 12(x2)3y3 ? y210"3 12x6y5

5"3 6x4y42"3 2x2y6!3 2"3 63 ? 2

!3 432!3 12 ? 36!3 36!3 12"202

!400!10 ? 40!40!10!2xy

"2 ? 42 ? x ? (x4)2 ? y ? (y2)2"32x9y5

"3 f(y 1 3)2g3"3 (y 1 3)6

"5 (3x3)5"5 243x15"f(a 2 1)2g2"(a 2 1)4"3 (24y3)3"3 264y9

"72x2(y5)2"49x2y10

2"3 332!3 27"0.52

!0.25"4 74"3 (20.4)3

!320.064"122!144

43. ? = =

= - 44. = 45. =

46. = = 47. 30.2= =

48. p-2.25= = = 49. =

4= 34

= 81 50. = ( )3= 63

= 216

51. = x1= x 52. ? = =

53. = x-6y4= 54. = -5;

3x + 1 = (-5)3; 3x + 1 = -125; 3x = -126; x = -4255. = x + 1; x + 7 = (x + 1)2; x + 7 =

x2+ 2x + 1; x2

+ x - 6 = 0; (x + 3)(x - 2) = 0; x =

-3 (extraneous) or x = 2 56. - 3 = 8; = 11;x = 121 57. ( f + g)(x) = f(x) + g(x) = 2x + 5 +

x2- 3x + 2 = x2

- x + 7 58. f(x) - g(x) = 2x +

5 - (x2- 3x + 2) = -x2

+ 5x + 3 59. g(x) ? f(x) =(x2

- 3x + 2)(2x + 5) = 2x3+ 5x2

- 6x2-

15x + 4x + 10 = 2x3- x2

- 11x + 1060. (g - f)(x) = g(x) - f(x) = x2

- 3x + 2 -

(2x + 5) = x2- 5x - 3 61. = ,

62. (g + f )(-x) = g( f(-2)) = g((-2)2) =

g(4) = 4 - 3 = 1 63. ( f + g)(-2) = f(g(-2)) =f(-2 - 3) = f(-5) = (-5)2

= 25 64. ( f + g)(0) =f(g(0)) = f(0 - 3) = f(-3) = (-3)2

= 9

65. (g + g)(7) = g(g(7)) = g(7 - 3) = g(4) =

4 - 3 = 1 66. ( f + g)(c) = f(g(c)) = f(c - 3) =

(c - 3)2= c2

- 6c + 9 67. y = 6x + 2;inverse: x = 6y + 2; 6y = x - 2; y = x - ; yes

68. y = 2x3+ 1; inverse: x = 2y3

+ 1; 2y3= x - 1;

y3= ; y = ; yes 69. y = (x - 2)4;

inverse: x = (y - 2)4; 4 = y - 2; y = 4 + 2;no 70. y = ; inverse: x = ; x2

= y +

2; y = x2- 2, x � 0; yes 71. (f -1 + f )(5) = 5

72. (f -1 + f )(-5) = -5 73. ( f-1 + f )(6) = 6

74. (f-1 + f )(t) = t

75. y = + 3 76. y =

77. y = 78. y = !x 1 7 2 22 3!x 1 6

!x 2 1!x

!y 1 2!x 1 2!4 x!4 x

Qx 2 12 R1

3x 2 12

13

16

x 2 252

x2 2 3x 1 22x 1 5

g(x)f(x)

!xx12

!x 1 7

!3 3x 1 1y4

x6Ax23

8y14B16

x56x

161

23x

23x

16Ax3

4B43!3636

32A!5 243B

24345Å

4 1p9

1"4 p9p2

94

!5 3315Å

4 18

1"4 2322

34"3 x2

x23!5 33

15!2 1 !10

4!2 1 !10

24

!2 1 !101 2 5

(1 1 !5)(1 1 !5)

!2(1 2 !5)

42 2 12!337


42Ox

y6

4

2 �4�8 O x

y

4

42Ox

y

4

2

42O x

y

4

2

CHAPTER TEST page 432

1. = -0.3 2. =

= 3�x � y2

3. =

-2x2y4 4. = (x - 2)2

5. =

6. =

= 6x3y4

7. = = ? = =

8.

9.

=

10.

11.

=

-17 - 4 12. (7 + )(3 + 5 ) = 21 + 35 +

= 13.

= = =

14. = =

15. = =

16. = 17. =

= = = =

18. ; x - 3 = (x - 5)2;

x - 3 = x2- 10x + 25; x2

- 11x + 28 = 0;(x - 7)(x - 4) = 0; x - 7 = 0 or x - 4 = 0; x = 7or x = 4 (extraneous); x = 7 19. ;x + 4 = 3x; 2x = 4; x = 2 20. ;

3x + 4 = (-5)3; 3x + 4 = -125; 3x = -129; x = -4321. x - 6 = ; (x - 6)2

= x - 4; x2- 12x +

36 = x - 4; x2- 13x + 40 = 0; (x - 8)(x - 5) = 0;

x - 8 = 0 or x - 5 = 0; x = 8 or x = 6 (extraneous);x = 8 22. g(x) - f(x) = x2

- 3x + 2 - (x - 2) =x2

- 4x + 4; domain: all real numbers 23. -2g(x) +f(x) = -2(x2

- 3x + 2) + x - 2 = -2x2+ 6x - 4 +

x - 2 = -2x2+ 7x - 6; domain: all real numbers

24. = = = x - 1;

domain: all real numbers x 2 2 25. -f (x) ? g(x) =-(x - 2)(x2

- 3x + 2) = (2 - x)(x2- 3x + 2) =

2x2- 6x + 4 - x3

+ 3x2- 2x = -x3

+ 5x2- 8x + 4;

domain: all real numbers 26. f (g(x)) = f (4x + 1) =

(x 2 2)(x 2 1)x 2 2

x2 2 3x 1 2x 2 2

g(x)f(x)

(x 2 4)12

(3x 1 4)13 5 25

!x 1 4 5 !3x

!x 2 3 5 x 2 54x4"3 x2

9y6

4"3 (x4)3 ? x2

9y64"3 x14

9y622x624

3

32y6(!3 8)2x6y2

(!3 27)2x43y8

a 8x9y3

27x2y12b23

x161

13 5 x

12x

16 ? x

131

52 51

25

1("3 125)2(125)2

233 1 !2 1 !3 1 !6

!3 1 !2 1 3 1 !63 2 2

1 1 !3(!3 2 !2)

?(!3 1 !2)(!3 1 !2)

21 2 23!615 1 10!6

21515 1 10!6

9 2 24

(3 1 2!6)(3 1 2!6)

5(3 2 2!6)

?36 1 38!33!3 1 15

!3!3!3!5

3 2 6!5 1 2!5 2 20 52!5 2 2!1003!20 1

(1 2 !20) 5 3 2(3 1 2!5)12!2 2 !5

7!2 1 5!2 2 !5 5!5 5!98 1 !50 210!3 1 10!3 5 24!34!3 15(2!3)

4!3 1 2(5!3) 15!12 52!75 1!48 1

!6x 2 3x!26x

!6x!6x

5 !6x 2 "18x2

6x 51 2 !3x!6x

?

x!xy3y

"x3y3y

!y!yÅ

x39yÅ

7x4y63xy2

"7x4y

"63xy2

!2y"62 ? 2 ? (x3)2 ? (y4)2 ? y

"72x6y9 5"3y3 ? "4xy4 ? "6x5y2

7x2!2"(7x2)2 ? 2 5"7x3 ? !14x 5 "98x4

"(x 2 2)4 5 C(x 2 2)4D12"5 2x4

"5 264x14y20 5 "5 (22)5 ? 2 ? (x2)5 ? x4 ? (y4)5

!6xy"32 ? 6 ? x2 ? x ? (y2)2 ? y

"54x3y5!320.027 5 "3 (20.3)3

(4x + 1)2- 2 = 16x2

+ 8x + 1 - 2 = 16x2+ 8x - 1;

g(f(x)) = g(x2- 2) = 4(x2

- 2) + 1 = 4x2- 8 + 1 =

4x2- 7 27. f (g(x)) = f(-3x - 1) = 2(-3x - 1)2

+

(-3x - 1) - 7 = 2(9x2+ 6x + 1) - 3x - 1 - 7 =

18x2+ 12x + 2 - 3x - 8 = 18x2

+ 9x - 6; g( f (x)) =g(2x2

+ x - 7) = -3(2x2+ x - 7) - 1 = -6x2

- 3x +

21 - 1 = -6x2- 3x + 20 28. The sixth power of a

real number is always nonnegative. 29a. f(x) = 0.5x29b. g(x) = 0.75x 29c. g(g(x)) = g(0.75x) =0.75(0.75x) = 0.5625x 29d. The cashier’s solution istoo high by 6.25% of the original price.30. y = - 1; y = - 1; y =

4 - 1; the graph of y = 4 - 1 is thegraph of y = 4 , translated 5 units left and 1 unit

down. 31. y = ; y = ; y =

; the graph of y = is the graph of y =

3 , translated unit left.

32. y = 2 + 3

domain: real numbers � 0;range: real numbers � 3

33. y =

domain: real numbers � - ;

range: real numbers � 0

34. y = -

domain: real numbers � 4;range: real numbers � 0

35. y = - 4

domain: real numbers � - ;

range: real numbers � -436. f (g(-2)) = f (7(-2) - 4) = f (-18) = (-18)3

+

1 = -5831 37. g(f(3)) = g(33+ 1) = g(28) =

7(28) - 4 = 192 38. f(g(0)) = f (7(0) - 4) =

f (-4) = (-4)3+ 1 = -63 39. f(x) = 3x3

- 2,

or y = 3x3- 2; inverse: x = 3y3

- 2; 3y3= x + 2;

y3= ; y = ; ; yes f21(x) 5 Å

3 x 1 23Å

3 x 1 23

x 1 23

32

!x 1 3

!x 2 412

32

!2x 1 3

!x

13!x

3Åx 1 133Åx 1 1

3

Å9Qx 1 13R!9x 1 3

!x!x 1 5!x 1 5

!16(x 1 5)!16x 1 80


Ox

y

4

2

2 4

�2 2O x

y

2

�2

2O x

y

2

�2

�4 2Ox

y2

�4

�8

40. g(x) = - 1, or y = - 1;

inverse: x = - 1; = x + 1; y + 3 =

(x + 1)2; y = (x + 1)2- 3; g-1(x) = (x + 1)2

- 3; yes

41. g(x) = , or y = ;

inverse: x = ; x2= 2y + 1; 2y = x2

- 1;

y = ; g-1(x) = ; yes 42. f (x) = x4, or

y = x4; inverse: x = y4; 4x = y4; y = 4 ;

f-1(x) = 4 ; no 43a. V = p ? 43= �

268.08; about 268.08 in.3 43b. V = pr3; 3V = 4pr3;

= r3; r = , or r = 43c. r = �

2.88; about 2.88 in. 44. Check students’ work.

45. t = 0.2 � 0.6; about 0.6 seconds;

t = 0.2 � 0.9; about 0.9 seconds

CA STANDARDS MASTERY pages 433–435

Vocabulary ReviewA. III; radicand: the number under the radical sign in aradical expression B. II; index: the degree of the rootin a radical expression C. I; composite function: thecombination of two functions such that the output fromthe first becomes the input for the second D. V;inverse function: the range of one function is thedomain of the other and vice versa E. IV; radicalfunction: a function that can be written in the form f(x)= a + k

Multiple Choice

1. = ; this is true for w = 1, but false for w

= –1; the answer is B. 2. (x + 1)2= x2

+ 1; substitute

0 for x: (0 + 1)2= 0 + 1 is a true statement; substitute 1

for x: (1 + 1)2= 12

+ 1 is not a true statement; the

answer is B. 3. y = = ; A: when x � 0 and n �

0, y � 0 is valid for real values of y; the answer is A. 4.

= = = ; the answer is

C. 5. = 27; x + 2 = ( ; x + 2 = 34; x +

2 = 81; x = 79; the answer is C. 6. A: = ( ; B:

= ( = 32= 9; the answer is B. 7.

= ) ; the answer is A. 8. A: From 0 to 12

months, the dotted line representing Elbac cable is

below the solid line representing StableCable; Elbac

would cost less than StableCable if Lucinda used fewer

than 12 channels per month; the answer is B.9. (1) 2x + 3y - z = -2, (2) x - 4y + 2z = 18, (3) 5x +2y - 6z = 8; multiply (1) by 2 and add (2) to eliminatez:

(b32)"5 (1296a8

(b3)12(6a2)

45"3 27)227

23

"3 90)29023

"3 27)4(x 1 2)34

(3xy2)43y

23(3x)

23"3 9x2y4"6 81x4y8

x1n"n x

(w3)13(w4)

14

"3 x 2 h

!20

!10

Q3(100)4p R

13Q3V

4p R13

Å3 3V

4p3V4p

43

256p3

43!4 4x

!4 4x14

14

14

x2 2 12

x2 2 12

!2y 1 1

!2x 1 1!2x 1 1

!y 1 3!y 1 3

!x 1 3!x 1 3 2x + 3y - z = -2 u 4x + 6y - 2z = -4 x - 4y + 2z = 18 u x - 4y + 2z = 18

(4) 5x + 2y = 14

Multiply (1) by -6 and add (3) to eliminate z:2x + 3y - z = -2 u -12x - 18y + 6z = 12 5x + 2y - 6z = 8 u 5x + 2y - 6z = 8

(5) -7x - 16y = 20

Multiply (4) by 8 and add equations (4) and (5) toeliminate y:

5x + 2y = 14 u 40x + 16y = 112-7x - 14y = 20 u -7x - 16y = 20

33x = 132 x = 4

Substitute 4 for x into (4) and solve for y: 5(4) + 2y =14; 20 + 2y = 14; 2y = -6; y = -3; substitute 4 for xand -3 for y into (2): (4) -4(-3) + 2z = 18; 4 + 12 +2z = 18; 16 + 2z = 18; 2z = 2; z = 1; the answer is C.10. x = 5y − 1; x + 1 = 5y; y = ; the answer is B.

11. Use composition of functions to check each answer:

A: f(x) = x - 4; (f o g)(x) = (x - 3) - 4 = x - 7; B:

f(x) = x - 1; (f o g)(x) = (x - 3) - 1 = x - 4; C: f(x) =

x2- 1; (f o g)(x) = (x - 3)2

- 1 = x2 − 6x + 9 - 1 = x2

− 6x + 9 + 8; the answer is C. 12. g(1) = (1) – 3 = -2;

h(-2) = (–2)2+ 6 = 4 + 6 = 10; the answer is D. 13.

The florist has = 11,440 ways to choose the 6

of the 21 carnations; the answer is C. 14. There are

= 55 ways to choose 2 of the 11 pens, and

there are = 210 ways to choose 4 of the 10

pencils; 55 � 210 = 265 ways to choose pens and

pencils for class; the answer is C. 15. (x3)4+ (x2)3

=

x12+ (x2)3

= x12+ x6; step 2 is incorrect; the answer is B.

16. = ? ? = x3 ? ? = x3 ?

y2 ? ; step 3 is incorrect; the answer is C.17. f(x) − g(x) = 3x2 − 5x + 3 − (x − 1)2

= 3x2 − 5x + 3− x2

+ 2x − 1 = 2x2 − 3x + 2; the answer is A.18. f(g(x)) = (2x − 3)2 − 1 = 4x2 − 12x + 9 − 1 = 4x2 −12x + 8; the answer is D. 19. (1) 2x − y = 4 and (2) y =2 − x; substitute 2 − x for y in (1) and solve for x; 2x − (2 −x) = 4; 3x − 2 = 4; 3x = 6; x = 2; substitute 3 for x in (2)and solve for y; y = 2 − (2) = 0; (x, y) = (2, 0); the answeris B. 20. 3 + (x − 1) = 30; (x − 1) = 27; x − 1 =( ; x − 1 = 34; x − 1 = 81; x = 82; the answer is D.21. Check each ordered pair; A: (3, 0): 2(3) − 0 � 3 and2(3) + 0 ≥ 5; 6 � 3 and 6 � 5; B: (6, 1): 2(6) − 1 � 3 and2(6) + 1 ≥ 5; 11 � 3 and 13 � 5; C: (4, −1): 2(4) − (−1) � 3and 2(4) + (−1) � 5; 9 � 3 and 7 � 5; D: (0, −3): 2(0) −(−3) � 3 and 2(0) + (−3) � 5; 5 � 3 and −1 � 3; (0, –3) isnot a solution of the system; the answer is D. 22. x = y2

− 3; x + 3 = y2; y = ± ; the answer is C."x 1 3

"3 27)4

34

34

"3 z"3 z"3 y6"3 z"3 y6"3 x9"3 x9y6z

10!(10 2 4)!4!

11!(11 2 2)!2!

16!(16 2 9)!9!

x 1 15


Documents

2 Solution Key • Chapter 7, page 190 CHECK YOUR READINESS page 372 1. (3y - 2)(y - 4) = 3y2 - 12y - 2y + 8 = 3y2 - 14y + 8 2. (7a + 10)(7a - 10) = 49a2 - 70a + 70a …