329 Ï

Chapter 6,
continued
n}
90. Sample answer: Ï xm
n 5 3, m 5 5, x 5 21:
n 5 4, m 5 5, x 5 21:
11. (x6y 4)1/8 1 2(x1/3y1/4)2 5 x3/4y1/2 1 2(x2/3y1/2)
}
(21) 5 21
Ï}
4}
4
, but 21 has no
Ï(21)5 5 Ï21}
3
5
real 4th root; Ï{(21) {5 1
5
4
n 5 3, m 5 4, x 5 21:
n 5 2, m 5 4, x 5 21:
}
(21)4 5 1
Ï3 }
Ï(21)4 5 1
}
}
}
7Ï7
Ï75
3Ï7 1 4Ï 7
}
12. }}
5
} + }
}
}
Ï75 Ï75
Ï75
3/2
7 + 7 + 75/2
71 1 3/2 1 5/2
75
5}
5 }5 5 1
5}
5
5
7
7
7
3
}
Absolute value is needed when n is even and m is odd.
3
}
3
}
}
Ïx
2Ïx 4
2x2
2x2Ï x
Ïx
2Ï x + Ïx 3
}
13. }
5
5 }6
} 5 }
}
} 5 }
} + }
7
8x8
4x
Ï82 + x14 + x 8x Ïx Ï x
Ï64x15
5}
}
}
}
5}
5}
91. A;
22 2 3
5}
5 24xy2Ï2x
5}
5 }8
slope 5 }
24 2 4
28
15. Three labels A, B, C, are added to the graph to indicate
Choice A: 25x 1 8y 5 14
the three right triangles.
8y 5 5x 1 14
5
7
a2 1 b2 5 c2
(8 2 2)2 1 82 5 c2
62 1 82 5 c2
100 5 c2
10 5 c
For right triangle B
a2 1 b2 5 c2
42 1 82 5 c2
80 5 c2
}
4Ï 5 5 c
5
slope 5 }8
92. G;
25 a 26x 1 3 a 15
26x 1 3 a 15
6x a 8
26x a 12
4
x a }3
x q 22
4
22 a x a }3
}
1. 363/2 5 (Ï 36 )3 5 63 5 216
1
1
1
1
2. 6422/3 5 }
5}
5}5}
}
642/3
(Ï3 64 )2 42 16
8
For right triangle C
a2 1 b2 5 c 2
22 1 42 5 c 2
20 5 c 2
}
2Ï 5 5 c
1. f(x) 1 g(x) 5 22x 2/3 1 7x2/3 5 (22 1 7)x 2/3 5 5x2/3
2. f(x) 2 g(x) 5 22x2/3 2 7x2/3 5 (22 2 7)x2/3 5 29x2/3
5}
4. (232)2/5 5 (Ï 232 )2 5 (22)2 5 4
5. x4 5 20
3. The functions f and g each have the same domain: all real
6. x5 5 210
4}
numbers. So, the domains of f 1 g and f 2 g also consist
of all real numbers.
4. f(x) + g(x) 5 3x(x1/5) 5 3x(1 1 1/5) 5 3x6/5
5}
x 5 6Ï 20
x 5 Ï 210
x ø 62.11
x ø 21.58
7. x6 1 5 5 26
3}
f (x)
3x
5. } 5 }
5 3x(1 2 1/5) 5 3x4/5
g(x)
x1/5
3}
6. The functions f and g each have the same domain: all real
8. (x 1 3)3 5 216
x6 5 21
x 1 3 5 Ï 216
x 5 6 Î21
6 }
x 5 Ï 216 2 3
x ø 61.66
x ø 25.52
4}
9. Ï 32 + Ï 8 5 Ï 32 + 8 5 Ï 256 5 4
}
4
6.3 Guided Practice (pp. 429–431)
3. 2(6253/4) 5 2[(Ï 625 )3] 5 2(53) 5 2125
4}
B
4
Lesson 6.3
4}
4}
A
The perimeter of the right triangle is
}
}
}
10 1 4Ï 5 1 2Ï5 5 10 1 6Ï 5 .
Quiz 6.1–6.2 (p. 427)
4}
2
C
For right triangle A
y 5 }8 x 1 }4
25 a 26x 1 3
5}
5 2xy2Ï 2x 2 6xy2Ï 2x
5
25
}
5
5
14. y 2Ï 64x6 2 6Ï
2x6y10 5 y2Ï 32 + 2x5x 2 6Ï
2x5xy10
Mixed Review for TAKS
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
5 x3/4y1/2 1 2x2/3y1/2
}
3}
10. (Ï 10 + Ï 10 )8 5 (101/2 + 101/3)8
5 F 10
G
(1/2 1 1/3) 8
5 (105/6)8
numbers. So, the domain of f + g also consists of all real
f
numbers. Because g(0) 5 0, the domain of }g is restricted
to all real numbers except x 5 0.
7. r(m) + s(m) 5 (1.446 3 109)m20.05
5 (1.446 3 109)(1.7 3 105)20.05
ø (1.446 3 109)(0.55)
20/3
5 10
3}
20
5 Ï 10
3}
5 Ï 1018 + 102
}
3
(106)3 + 102
5Ï
3}
5 106Ï 102
3}
5 1,000,000Ï 100
ø 791,855,335
The white rhino has about 791,855,335 heartbeats over
its lifetime.
8. g( f (5))
9. f ( g(5))
f(5) 5 3(5) 2 8 5 7
g(5) 5 2(5)2 5 50
g( f (5)) 5 g(7)
f ( g(5)) 5 f (50)
5 2(7)2
5 3(50) 2 8
5 98
5 142
Algebra 2
Worked-Out Solution Key
329
Chapter 6,
continued
6. g(x) 1 g(x) 5 5x1/3 1 4x1/2 1 5x1/3 1 4x1/2
11. g( g (5))
2
g(5) 5 2(5) 5 50
g( g(5)) 5 g(50)
5 2(50)2
5 5000
f (5) 5 3(5) 2 8 5 7
f ( f (5)) 5 f (7)
5 3(7) 2 8
5 13
12. f (x) 5 2x21, g(x) 5 2x 1 7
21
f (g(x)) 5 f (2x 1 7) 5 2(2x 1 7)
5 (5 1 5)x1/3 1 (4 1 4)x1/2
5 10x1/3 1 8x1/2
Domain: all nonnegative real numbers
7. f(x) 2 g(x) 5 23x1/3 1 4x1/2 2 (5x1/3 1 4x1/2)
5 (23 2 5)x1/3 1 (4 2 4)x1/2
2
5}
2x 1 7
4
g( f (x)) 5 g (2x21) 5 2(2x21) 1 7 5 4x21 1 7 5 }x 1 7
f( f (x)) 5 f (2x21) 5 2(2x21)21 5 2(221x) 5 20x 5 x
5 28x1/3
Domain: all nonnegative real numbers
8. g(x) 2 f (x) 5 5x1/3 1 4x1/2 2 (23x1/3 1 4x1/2)
5 (5 1 3)x1/3 1 (4 2 4)x1/2
The domain of f( g(x)) consists of all real numbers except
7
5 8x1/3
7
x 5 2}2 because g1 2}2 2 5 0 is not in the domain of f.
The domains of g( f (x)) and f ( f (x)) consist of all real
numbers except x 5 0, again because 0 is not in the
domain of f.
Domain: all nonnegative real numbers
9. f(x) 2 f(x) 5 23x1/3 1 4x1/2 2 (23x1/3 1 4x1/2)
5 (23 1 3)x1/3 1 (4 2 4)x1/2
13. Function for $15 gift certificate: f (x) 5 x 2 15
Function for 20% discount: g(x) 5 x 2 0.2x 5 0.8x
g( f (x)) 5 g(x 2 15) 5 0.8(x 2 15)
f (g(x)) 5 f (0.8x) 5 0.8x 2 15
When x 5 55:
g( f (55)) 5 0.8(55 2 15) 5 0.8(40) 5 $32
f (g(55)) 5 0.8(55) 2 15 5 44 2 15 5 $29
The sale price is $32 when the $15 gift certificate is
applied before the 20% discount. The sale price is $29
when the 20% discount is applied before the $15 gift
certificate.
10. g(x) 2 g(x) 5 5x1/3 1 4x1/2 2 (5x1/3 1 4x1/2)
5 (5 2 5)x1/3 1 (4 2 4)x1/2
50
Domain: all nonnegative real numbers
11. B;
f(x) 1 g(x) 5 27x 2/3 2 1 1 2x2/3 1 6
5 (27 1 2)x 2/3 2 1 1 6
5 25x 2/3 1 5
12. f(x) + g(x) 5 4x2/3 + 5x1/2
6.3 Exercises (pp. 432–434)
5 20x(2/3 1 1/2)
Skill Practice
1. The function h(x) 5 g( f (x)) is called the composition of
the function g with the function f.
2. The sum of two power functions is sometimes a power
function.
5 20x7/16
Domain of f: all real numbers
Domain of g: all nonnegative real numbers
Domain of f + g: all nonnegative real numbers
13. g(x) + f(x) 5 5x1/2 + 4x2/3
Sample answer:
f (x) 5 2x1/3, g(x) 5 4x21/3
5 20x(1/2 1 2/3)
f (x) 1 g(x) 5 2x1/3 1 4x21/3
5 20x7/6
Domain of g: all nonnegative real numbers
f (x) 5 2x1/3, g(x) 5 4x1/3
1/3
f (x) 1 g(x) 5 2x
1/3
1 4x
1/3
3. f (x) 1 g(x) 5 23x
5 6x
1 4x
5 (23 1 5)x
Domain of f: all real numbers
1/3
1/2
1/3
1/3
1/2
1 5x
1 4x
1/2
1 (4 1 4)x
4. g(x) 1 f(x) 5 5x1/3 1 4x1/2 2 3x1/3 1 4x1/2
5 (5 2 3)x
5 2x
1/3
1/2
1 (4 1 4)x
1 8x
5. f (x) 1 f(x) 5 23x1/3 1 4x1/2 1 (23x1/3 1 4x1/2)
1/3
5 26x
1/2
1 (4 1 4)x
1/2
1 8x
Domain: all nonnegative real numbers
Algebra 2
Worked-Out Solution Key
Domain of f + f: all real numbers
5 25x(1/2 1 1/2)
Domain: all nonnegative real numbers
5 (23 2 3)x
Domain of f: all real numbers
15. g(x) + g(x) 5 5x1/2 + 5x1/2
1/2
1/3
14. f(x) + f(x) 5 4x2/3 + 4x2/3
5 16x4/3
Domain: all nonnegative real numbers
1/3
Domain of g + f: all nonnegative real numbers
5 16x(2/3 1 2/3)
5 2x1/3 1 8x1/2
330
50
Domain: all nonnegative real numbers
5 25x
Domain of g: all nonnegative real numbers
Domain of g + g: all nonnegative real numbers
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
10. f ( f (5))
Chapter 6,
continued
f (x)
4x2/3
4x(2/3 2 1/2)
4x1/6
16. } 5 }
5}
5}
5
5
g(x)
5x1/2
3x21 1 4
3x21
4
1
4
30. h( f (x)) 5 h(3x21) 5 } 5 } 1 } 5 }
x1}
3
3
3
3
Domain of f: all real numbers
Domain of g: all nonnegative real numbers
f
Domain of }g: all positive real numbers
g(x)
5x1/2
5x(1/2 2 2/3)
5x21/6
5
17. } 5 }
5}
5}
5}
4
4
f (x)
4x2/3
4x1/6
Domain of f: all real numbers
Domain of g: all nonnegative real numbers
g
f
Domain of }: all positive real numbers
f (x)
4x2/3
18. } 5 }
51
f (x)
4x2/3
2
1
2
2x 1 8
2x 1 8 2 21
2x 2 13
275}
5}
5}
3
3
3
The domain of g(h(x)) consists of all real numbers.
2x 2 7 1 4
2x 2 3
32. h(g(x)) 5 h(2x 2 7) 5 } 5 }
3
3
The domain of h(g(x)) consists of all real numbers.
33. f( f(x)) 5 f (3x21) 5 3(3x21)21 5 3(321x) 5 30x 5 x
x14
f
f
Domain of }: all real numbers except x 5 0
5x1/2
5x
}14
x 1 16
3
x 1 4 1 12
x14
34. h(h(x)) 5 h } 5 } 5 } 5 }
3
9
9
3
1
2
The domain of h(h(x)) consists of all real numbers.
35. g(g(x)) 5 g(2x 2 7) 5 2(2x 2 7) 2 7
19. } 5 }
1/2 5 1
Domain of g: all nonnegative real numbers
g
Domain of }g: all positive real numbers
2
20. g(23) 5 2(23) 5 29
f (g(23)) 5 f (29) 5 3(29) 1 2 5 225
21. f (2) 5 3(2) 1 2 5 8
225 2 2
27
h( f (29)) 5 h(225) 5 }
5 2}
5
5
822
6
23. h(8) 5 } 5 }
5
5
6 2
36
6
g(h(8)) 5 g }5 5 2 }5 5 2}
25
1 2
24. g(5) 5 252 5 225
225 2 2
27
5 2}
h(g(5)) 5 h(225) 5 }
5
5
for x in the function f.
f (g(x)) 5 f (4x) 5 (4x)2 2 3 5 16x2 2 3
37. The product 4(x2 2 3) was not performed correctly.
3
49x
g( f (x)) 5 g(7x2) 5 3(7x 2)22 5 3(722x24) 5 }4
39. Sample answer: f (x) 5 x, g(x) 5 x21
f (g(x)) 5 g( f (x))
f (x21) 5 g(x)
x21 5 x21
40. Sample answer:
3}
f (x) 5 Ï x , g(x) 5 x 1 2
3}
25. f (7) 5 3(7) 1 2 5 23
h(x) 5 f (g(x)) 5 f (x 1 2) 5 Ï x 1 2
f ( f (7)) 5 f (23) 5 3(23) 1 2 5 71
6
24 2 2
26. h(24) 5 } 5 2}
5
5
6
}22
2
6
5
36. When performing f (4x), 4x should have been substituted
38. A;
22. f (29) 5 3(29) 1 2 5 225
1 2
5 4x 2 14 2 7 5 4x 2 21
The domain of g(g(x)) consists of all real numbers.
g( f (x)) 5 g(x 2 2 3) 5 4(x 2 2 3) 5 4x 2 2 12
g( f (2)) 5 g(8) 5 282 5 264
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
1
x14
x14
31. g(h(x)) 5 g } 5 2 } 2 7
3
3
The domain of f ( f (x)) consists of all real numbers except
x 5 0, because 0 is not in the domain of f.
Domain of f: all real numbers
g(x)
g(x)
The domain of h( f (x)) consists of all real numbers
except x 5 0 because 0 is not in the domain of f.
41. Sample answer:
4
f (x) 5 }
, g(x) 5 3x 2
x17
5 2}
h(h(24)) 5 h1 2}5 2 5 }
5
25
16
27. g(25) 5 2(25)2 5 225
g(g(25)) 5 g(225) 5 2(225)2 5 2625
3
28. f (g(x)) 5 f (2x 2 7) 5 3(2x 2 7)21 5 }
2x 2 7
4
3x 1 7
h(x) 5 f (g(x)) 5 f (3x2) 5 }
2
42. Sample answer:
f (x) 5 {x{, g(x) 5 2x 1 9
h(x) 5 f (g(x)) 5 f (2x 1 9) 5 {2x 1 9{
The domain of f(g(x)) consists of all real numbers except
7
7
x 5 }2 because g1 }2 2 5 0 is not in the domain of f.
6
29. g( f (x)) 5 g(3x21) 5 2(3x21) 2 7 5 6x21 2 7 5 }
x27
The domain of g( f(x)) consists of all real numbers
except x 5 0 because 0 is not in the domain of f.
Algebra 2
Worked-Out Solution Key
331
Chapter 6,
continued
}
Problem Solving
Ïx 2 1 144
20 2 x
b. t(x) 5 r(x) 1 s(x) 5 } 1 }
6.4
0.9
1.1w 0.734
43. r(w) 5 }
b(w) 2 d(w)
c.
1.1w 0.734
5 }}
0.007w 2 0.002w
1.1w 0.734
5}
0.005w
Minimum
X=1.7044344 Y=16.325839
5 220w(0.734 2 1)
The value of x that minimizes t(x) is 1.7. This means
that to get to the ball in the shortest time, Elvis should
run along the beach 20 2 1.7 5 18.3 meters and then
swim out to the ball.
5 220w20.266
r(w) 5 220w20.266
2
1 1 }1
3
47. a. f(1) 5 } 5 } 5 1.5
2
2
2
1.5 1 }
1.5
ø 1.417
f ( f (1)) 5 f (1.5) 5 }
2
2
1.417 1 }
1.417
ø 1.414
f ( f ( f (1))) 5 f (1.417) 5 }}
2
f( f ( f ( f (1)))) 5 f (1.414)
2
1.414 1 }
1.414
44. C(x(t)) 5 C(50t) 5 60(50t) 1 750 5 3000t 1 750
ø 1.414214
5 }}
2
C(x(5)) 5 3000(5) 1 750 5 15,750
b. f ( f ( f ( f ( f (1))))) 5 f(1.414214)
This number represents the cost ($15,750) of 5 hours of
production in the factory.
2
1.414214 1 }
1.414214
}}
ø 1.414214
5
45. Let x represent the regular price.
2
}
Function for $15 discount: f (x) 5 x 2 15
Function for 10% discount: g(x) 5 x 2 0.1x 5 0.9x
Ï 2 ø 1.414213562
You need to compose the function
3 times in order for
}
the result to approximate Ï2 to three decimal places.
You need to compose the function
4 times in order for
}
the result to approximate Ï2 to six decimal places.
a. g( f (x)) 5 g(x 2 15) 5 0.9(x 2 15)
g( f (85)) 5 0.9(85 2 15) 5 0.9(70) 5 63
The sale price is $63 when the $15 discount is applied
before the 10% discount.
b. f (g(x)) 5 f (0.9x) 5 0.9x 2 15
f (g(85)) 5 0.9(85) 2 15 5 76.5 2 15 5 61.50
Mixed Review for TAKS
48. C;
(6x 3y 5z21)(23x24y 2) 5 218x21y 7z21
The sale price is $61.50 when the 10% discount is
applied before the $15 discount.
c. If the 10% discount is applied before the $15 discount,
you get a better deal. Your purchase will be $61.50
instead of $63.
18y7
5 2}
xz
49. G;
s 5 (100% 2 68%)t
s 5 32%t
46. a. Distance from point A to point D: 20 2 x
32
Distance 5 rate + time
t
s5}
100
20 2 x 5 (6.4)r(x)
20 2 x
6.4
8
s5}
t
25
} 5 r (x)
Distance from point D to point B:
x 2 1 122 5 c2
2
2
x 1 144 5 c
}
Ïx2 1 144 5 c
Distance 5 rate + time
}
Ïx 2 1 144 5 (0.9)s(x)
}
Ïx 2 1 144
} 5 s(x)
0.9
332
Algebra 2
Worked-Out Solution Key
Graphing Calculator Activity 6.3 (p. 435)
1.
Y1=X3+5X-3
Y2=-3X2-X
Y3=Y2+Y1
Y4=
Y5=
Y6=
Y7=
Y7=
Y3(7)
221
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
r(6.5) 5 220(6.5)20.266 ø 134
The breathing rate of a mammal that weighs 6.5 grams is
about 134 breaths per minute.
r(300) 5 220(300)20.266 ø 48.3
The breathing rate of a mammal that weighs 300 grams is
about 48.3 breaths per minute.
r(70,000) 5 220(70,000)20.266 ø 11.3
The breathing rate of a mammal that weighs 70,000
grams is about 11.3 breaths per minute.
Chapter 6,
continued
2.
6. G;
Y1=X^(1/3)
Y2=9X
Y3=Y1/Y2
Y4=
Y5=
Y6=
Y7=
Y7=
Y3-8
()
277777778
0
.
0
16
4
5 1 }2 2
1}
4 2
1/2 5
5
1/2
5 25
5 32
7.
3.
Y1=5X3-3X2
Y2=-2X2-5
Y3=Y2-Y1
Y4=
Y5=
Y6=
Y7=
Y7=
Y3(2)
4
V 5 }3 :r 3
4
900 5 }3 (3.14)r 3
215 ø r 3
5.99 ø r
The radius of the sphere is about 5.99 inches.
4.
Y1=2X2+7X-2
Y2=X-6
Y3=Y1(Y2)
Y4=
Y5=
Y6=
Y7=
Y7=
Y3(5)
-7
Lesson 6.4
Investigating Algebra Activity 6.4 (p. 437)
1. a. f(x) 5 3x 1 2
Mixed Review for TEKS (p. 436)
1. A;
3/2
79
3Ï 4:F
V5}
}
21
0
1
2
y 5 f (x)
24
21
2
5
8
y
120
522
1
3
6
m5}5}
r(x) 5 x 2 2 }2 1 }2 x 2(x)
1 1
1
r(x) 5 x2 2 }4 x 2
3
r(x) 5 }4 x 2
128(8.8) 5 3.14(r 2)(2.5)
1126.4 ø 7.85r 2
143.5 ø r 2
611.98 ø r
The radius is about 12 feet.
f
2. a. You can graph the inverse of a function by reflecting it
in the line y 5 x.
3. a. In words, g is the function that subtracts 2 from x then
divides the result by 3.
f (g(x)) 5 f 1 }
3 2
x22
g( f (x)) 5 g(3x 1 2)
3x 1 2 2 2
12
5 31 }
3 2
5}
3
5x2212
5}
3
5x
5x
x22
g(f (x)) represents your bonus when x > 100,000.
V 5 :r 2h
(21, 21)
g(x) 5 }
3
3. D;
4. H;
x
x22
The volume is about 66 cubic inches.
r(x) 5 Area of square 2 Area of triangle
(8, 2)
g
(2, 0) 6
(22, 24)
x22
y5}
3
2. H;
(5, 1)
(24, 22)
1
y 2 0 5 }3 (x 2 2)
(2, 8)
(1, 5)
(0, 2)
y 2 y1 5 m(x 2 x1)
V ø 66.03
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
22
(2, 0), (5, 1)
V 5 321(4:)21/2(S 3)1/2
S 3/2
V5 }
}
3Ï 4:F
x
3x
If f (g(x)) 5 x and g( f (x)) 5 x, then the function is
indeed the inverse of the original function.
x21
1. b. f(x) 5 }
6
x
25
22
y 5 f(x)
21
2}2
1
1
4
7
0
}
1
2
1
5. C;
f (x) 5 5 2 x
f ( f (x)) 5 5 2 (5 2 x)
f( f (x)) 5 x
Algebra 2
Worked-Out Solution Key
333