P. 442-445

P. 442-445
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
Chapter 6,
continued
(0, 1), (1, 7)
6.4 Guided Practice (pp. 439– 442)
y
(1, 7)
6
1
( 2, 4)
1
(25, 21)
(4, 2 ) (7, 1)
(1, 0)
g
x
(1, 0) 6
y 2 y1 5 m(x 2 x1)
1.
y5x14
x5y14
1
y 2 1 5 6(x 2 0)
x245y
(22, 2 2 )
f 21(x) 5 x 2 4
1
(2 2 , 22)
y 2 1 5 6x
f(x) 5 x 1 4
(21, 25)
f
f ( f 21(x)) 5 f (x 2 4)
y 5 6x 1 1
g(x) 5 6x 1 1
2. b. You can graph the inverse of a function by reflecting it
2.
x 5 2y 2 1
x21
6x 1 1 2 1
1
2
1
2
}x 1 } 5 y
5 61 }
11
6 2
x21
6x
5}
6
5x2111
5x
5x
1
3
y 5 f (x)
(4, 0), (1, 2)
22
0
2
7
4
1
f ( f 21(x)) 5 f 1 }2x 1 }2 2
220
2
y 2 y1 5 m(x 2 x1)
2
y 2 0 5 2}3 (x 2 4)
22(x 2 4)
(25, 6)
(22, 4)
6
5x1121
5 x 2 }2 1 }2
5x
5x
1
1
1
1
f(x) 5 23x 1 1
x 5 23y 1 1
x 2 1 5 23y
(7, 22)
1
1
2}3 x 1 }3 5 y
(4, 0)
x
1
1
1
f 21(x) 52}3 x 1 }3
(6, 25)
y5}
3
f ( f 21(x)) 5 f 1 2}3x 1 }3 2
1
f
22x 1 8
1
5 231 2}3x 1 }3 2 1 1
1
y5}
3
22x 1 8
g(x) 5 }
3
1
5x2111
5x
2. c. You can graph the inverse of a function by reflecting it
in the line y 5 x.
f 21( f(x)) 5 f 21(23x 1 1)
3. c. In words, g is the function that multiplies x by 22 and
1
f (g(x)) 5 f 1 }
2
3
22x 1 8
3
3
3 22x 1 8
5 4 2 }2 1}
2
3
5 }}
3
5 4 2 1}
2
2
5}
3
5 4 2 (2x 1 4)
5}
3
541x24
5x
22x 1 8
3x
If f (g(x)) 5 x and g( f (x)) 5 x, then the function is
indeed the inverse of the original function.
1
5x
8
40
8
40
104
40
144
4. L 5 }R 1 } 5 }(13) 1 } 5 } 1 } 5 } 5 48
3
3
3
3
3
3
3
The band provides 13 pounds of resistance when it is
stretched to 48 inches.
28 1 3x 1 8
5x
Algebra 2
Worked-Out Solution Key
1
5 x 2 }3 1 }3
g( f (x)) 5 g1 4 2 }2 x 2
221 4 2 }2 x 2 1 8
1
5 2}3 (23x 1 1) 1 }3
then adds 8, dividing the result by 3.
334
5 }2(2x 2 1) 1 }2
1
y 5 23x 1 1
1
(4, 22)
3.
f 21( f(x)) 5 f 21(2x 2 1)
5 21 }2x 1 }2 2 2 1
1
(22, 7)
(0, 4)
(1, 2)
(2, 1)
1
1
22 25
y
g
m5}
5 2}3
124
4
1
f 21(x) 5 }2 x 1 }2
If f (g(x)) 5 x and g( f (x)) 5 x, then the function is
indeed the inverse of the original function.
1. c. f(x) 5 4 2 } x x
2
5x
x 1 1 5 2y
g( f (x)) 5 g1 }
6 2
5}
6
5x1424
5x
y 5 2x 2 1
3. b. In words, g is the function that multiplies x by 6 and
f (g(x)) 5 f (6x 1 1)
5x2414
f(x) 5 2x 2 1
in the line y 5 x.
then adds 1.
f 21( f (x)) 5 f 21(x 1 4)
5.
f(x) 5 x 6, xq0
f(x) 5 x
y
6
y 5 x6
y 5 x6
6
x 5 y6
y5 x
1
6}
6Ï x 5 y
6}
f 21(x) 5 Ï x
21
x
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
721
m5}
56
120
Chapter 6,
continued
1
g(x) 5 }
x3
27
6.
y
1
y5}
x3
27
1
P
10.7
} 5 t 0.272
1
1/0.272
x
21
x5}
y3
27
P 5 10.7t 0.272
11.
y
1
5 27 x 3
P
1}
10.7 2
3
y53 x
27x 5 y 3
3.68
P
1}
10.7 2
3}
Ï27x 5 y
3}
3Ï x 5 y
3}
g21(x) 5 3Ïx
25 3.68
64
2
64
53
y 5 24 x
x
21
64
x 5 2}
y3
125
64
y 5 2125 x 3
6.4 Exercises (pp. 442– 445)
Skill Practice
1. An inverse relation interchanges the input and output
values of the original relation.
125
2}
x 5 y3
64
2. A function g is an inverse of f provided f (g(x)) 5 x and
Î2125
x5y
64
}
g( f (x)) 5 x.
}
3.
3}
Ï2125x
Ï64
3}
25Ï x
4
3}
25Ï x
5.
f (x) 5 2x 3 1 4
y 5 2x 1 4
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Ï4 2 x 5 y
3}
f 21(x) 5 Ï4 2 x
9.
f (x) 5 2x 5 1 3
y 5 2x 5 1 3
x 5 2y 5 1 3
x 2 3 5 2y5
21
3
2
22
1
2
11.
7
}
3
7
3
7
3
x 2 }5 5 2}5y
5
y
12. In the last step, each term was not divided by 6.
y 5 27x 5 1 7
y 5 6x 2 11
x 5 6y 2 11
x 1 11 5 6y
x 2 7 5 27y 5
x
1 2 }7 5 y5
2
y55
}
x
5
1 2 }7 5 y
x27
27
x
x27
Î1 2 }7x 5 Î}
27
7
2}3x 1 }3 5 y
y 5 27x 1 7
x 5 27y5 1 7
5
2}2x 1 3 5 y
y 5 2}5 x 1 }5
Î12x 2 32
5
g21(x) 5
3
1
15
1
5
x 5 2}5 y 1 }5
g(x) 5 27x5 1 7
Î
2
x 2 2 5 2}3 y
}x 2 } 5 y
x
5x23
}
10.
2
x 5 2}3y 1 2
x 2 }3 5 5y
y5
}
y 5 2}3 x 1 2
1
1
5
2
10.
x 5 5y 1 }3
3
1
}x 2 } 5 y
2
2
5
1
y 5 5x 1 }3
y 5 2x 5 1 3
y 5 218x 2 5
x 5 218y 2 5
x 1 5 5 218y
2}
x2}
5y
18
18
1
9.
y
}
f 21(x) 5
8.
7
12
1
12
14
5
1
10
}x 2 } 5 y
}x 2 } 5 y 5
Î
x
y 5 10x 2 28
x 5 10y 2 28
x 1 28 5 10y
}x 1 } 5 y
y 5 12x 1 7
x 5 12y 1 7
x 2 7 5 12y
y 5 4 2x
1
4 2 x 5 y3
5
7.
3
3}
6
7
1
7
x 5 2y 1 4
x 2 4 5 2y 3
6.
}x 1 } 5 y
y 5 2x 3 1 4
3
5
2}2x 1 }2 5 y
y 5 7x 2 6
x 5 7y 2 6
x 1 6 5 7y
y
3
y 5 22x 1 5
x 5 22y 1 5
x 2 5 5 22y
1
}x 1 } 5 y
f 21(x) 5 }
4
1
2
1
4
1
4
}5y
4.
y 5 4x 2 1
x 5 4y 2 1
x 1 1 5 4y
}
5y
3}
8.
ø 22.7
You can predict that the average ticket price will reach
$25 about 23 years after 1995, or in 2018.
y
y 5 2}
x3
125
3
5t
When P 5 25: t 5 1 }
10.7 2
f (x) 5 2}
x3
125
7.
5 (t 0.272)1/0.272
2
x
6
11
6
}1}5y
}
}
5
Algebra 2
Worked-Out Solution Key
335
Chapter 6,
continued
13. In the first step, the terms involving x and y were
switched instead of just the variables x and y.
y 5 2x 1 3
x 5 2y 1 3
x 2 3 5 2y
32x5y
14. Sample answer:
F x12 G
2
5 5F 1 x 1 5 2 G 2 2
f (g(x)) 5 f 1 }
5 2
1/2
}
5x1222
5 (x2)1/2
5x
5x
2
g( f (x)) 5 g(x 1 4)
2
5x2414
5x1424
5x
5x
2
x 1 4 5 }3 y
3
3
3
2
}x 1 6 5 y
g( f (x)) 5 g(2x 1 3)
5 21 }2x 2 }2 2 1 3
5 }2(2x 1 3) 2 }2
5x2313
5 x 1 }2 2 }2
3
3
1
3
3
2
} x 1 6 5 g(x)
22. f(x) 5 x 7
x 5 y7
Ïx 5 y
Î
7}
f 21(x) 5 Ï x
g( f (x)) 5 g1 }4x3 2
Îx + 4
}
1
f (g(x)) 5 f [(4x)1/3]
F
1
5 41 }4x3 2
5 }4 (4x)
1
5 (x 3)1/3
5x
5x
5 }4 F (4x)1/3 G3
1
5y
64}
4+4
G
4}
Ï4x
1/3
5y
6}
2
4}
Ï4x
f 21(x)5 }
2
f (g(x)) 5 f (5x 1 5)
y 5 210x 6
x 5 210y 6
g( f (x)) 5 g1 }5x 2 1 2
1
1
5 51 }5x 2 1 2 1 5
1
5 }5 (5x 1 5) 2 1
5x1121
5x2515
5x
5x
x + 10,000
6}
Ï2100,000x
5y
6}
10
6}
Ï2100,000x
f 21(x) 5 2}
10
g( f (x)) 5 g(4x 1 9)
5 }4 (4x 1 9) 2 }4
5x2919
5 x 1 }4 2 }4
5x
5x
Algebra 2
Worked-Out Solution Key
Î
5y
6 6 2}
10 + 100,000
5 41 }4x 2 }4 2 1 9
9
Î
x
2}
5 y6
10
}
x
6 6 2}
5y
10
}}
9
1
f (x) 5 4x 1 9, g(x) 5 }4 x 2 }4
2
f(x) 5 210x 6, xa0
24.
1
f (x) 5 }5 x 2 1, g(x) 5 5x 1 5
1
x 5 4y 4
x
} 5 y4
4
}
x
6 4 }4 5 y
7}
5x
1
y 5 4x 4
y5x
3
1
f (x) 5 }4 x3, g(x) 5 (4x)1/3
9
1
f (g(x)) 5 f }4x 2 }4
23. f(x) 5 4x 4, xq0
7
9
1
9
9
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
1
x 5 }3 y 2 4
f (x) 5 2x 1 3, g(x) 5 }2 x 2 }2
5x
336
2 1/2
5 }
5
y 5 }3 x 2 4
f (x) 5 x 1 4, g(x) 5 x 2 4
1
19.
1 5x 2
5 51 }
22
5 2
2
1
18.
2
2 2 1 2 1/2
m 5 }3, b 5 24
1
1
f (x) 5 }3x 1 }3
f (g(x)) 5 f 1 }2x 2 }2 2
17.
2
5}
5
y 5 mx 1 b
1
3
f (g(x)) 5 f (x 2 4)
16.
1 5x
1/2 2
21. B;
}x 1 } 5 y
15.
g( f (x)) 5 g(5x2 2 2)
x12
f 21(x) 5 3x 2 1
y 5 3x 2 1
x 5 3y 2 1
x 1 1 5 3y
1
3
x 1 2 1/2
f(x) 5 5x 2 2 2, xq0; g(x) 5 1 }
5 2
20.
Chapter 6,
continued
f (x) 5 32x5
25.
y 5 32x
30.
y
1
5
1 x
x 5 32y 5
x
32
} 5 y5
Î
f(x) 5 2x 2 5
}
5 x
}5y
32
Because no horizontal line intersects the graph of f more
than once, the inverse of f is a function.
5}
Ïx
Ï32
5}
Ïx
}5y
2
}
5} 5 y
31.
y
5}
Ïx
1
f 21(x) 5 }
2
2
f (x) 5 2}5x 3
26.
27.
y5}
x2
25
2
x5}
y2
25
16
x 5 2}5 y 3
5
y
x
f(x) 5 26x 2
Î25
}
}
6 }
x5y
16
}
Î25x2 + +4 4 5 y
}
5
}
6}4Ï x 5 y
}
3}
Ï220x
}5y
2
5 }
2}4Ï x 5 y
3}
f
1
1
}x 5 y 2
Î25x2 5 y
3
32.
25
16
2}2 x 5 y 3
3
Because a horizontal line intersects the graph of f more
than once, the inverse of f is not a function.
16
2
y 5 2}5 x 3
4 x
1
f(x) 5 4 x 2 2 1
16 2
f(x) 5 }
x , xa0
25
Ï220x
(x) 5 }
2
21
f
Because no horizontal line intersects the graph of f more
than once, the inverse of f is a function.
33.
y
5 }
(x) 5 2}4Ï x
21
1
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
28. C;
1
1
y 5 2}
x3
64
1 3
x 5 2}
y
64
x
f(x) 5 3x 3
1 3
f (x) 5 2}
x
64
Because no horizontal line intersects the graph of f more
than once, the inverse of f is a function.
34.
y
264x 5 y 3
1
3}
Ï264x 5 y
3}
24Ï x 5 y
3}
24Ï x 5 g(x)
29.
x
21
f(x) 5 x 3 2 2
y
Because no horizontal line intersects the graph of f more
than once, the inverse of f is a function.
1
1 x
f(x) 5 3x 1 1
35.
y
2
2
Because no horizontal line intersects the graph of f more
than once, the inverse of f is a function.
x
f(x) 5 (x 2 4)(x 1 1)
Because a horizontal line intersects the graph of f more
than once, the inverse of f is not a function.
Algebra 2
Worked-Out Solution Key
337
Chapter 6,
36.
continued
2
f(x) 5 2}5x 6 1 8, xa0
41.
y
2
y 5 2}5x 6 1 8
2
x 5 2}5y 6 1 8
f(x) 5 uxu 1 4
2
2
x 2 8 5 2}5y 6
x
2
Because a horizontal line intersects the graph of f more
than once, the inverse of f is not a function.
37.
5
Î
}
y
2
2}2x 1 20 5 y 6
x
2
5
6 6 2}2x 1 20 5 y
Î
}
5
f 21(x) 5 2 6 2}2x 1 20
2x 3 2 6
42.
f(x) 5 4x 4 2 5x 2 2 6
f(x) 5 }
9
2x 3 2 6
y5}
9
Because a horizontal line intersects the graph of f more
than once, the inverse of f is not a function.
3
f (x) 5 }2 x 4, xq0
3
y 5 }2 x4
39.
f (x) 5 x 3 2 2
9x 5 2y 3 2 6
3
3
x 5 }2 y4
y5x 22
9x 1 6 5 2y 3
x 5 y3 2 2
}x 1 3 5 y 3
9
2
Î92x 1 3 5 y
9
f (x) 5 Î 2x 1 3
}
2
3
}x 5 y4
x 1 2 5 y3
Î2x
}
3
}
3}
Ïx 1 2 5 y
64}
5y
3
Î2x + 27
}
3}
f 21(x) 5 Ï x 1 2
5y
64}
3 + 27
}
21
43.
y 5 x4 2 9
x 5 y4 2 9
x 1 9 5 y4
4}
f
Ï54x
(x) 5 }
3
21
4}
6Ïx 1 9 5 y
3
f (x) 5 }4x 5 1 5
40.
}
f(x) 5 x 4 2 9, xq0
4}
Ï54x
5y
6}
3
3
4}
f 21(x) 5 Ïx 1 9
44.a. False
3
y
y 5 }4 x 5 1 5
3
x 5 }4 y5 1 5
3
1
x 2 5 5 }4 y5
4
3
f(x) 5
20
3
x2
1
x
}x 2 } 5 y5
Î
Because a horizontal line intersects the graph of f
more than once, the inverse of f is not a function.
}
5
4
3
20
3
}x 2 } 5 y
Î43x 2 203
}
f 21(x) 5
5
}
b. True
}
y
f(x) 5 x 3
1
1
x
Because no horizontal line intersects the graph of f
more than once, the inverse of f is a function.
338
Algebra 2
Worked-Out Solution Key
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
38.
2y 3 2 6
x5}
9
Chapter 6,
45.
continued
f (x) 5 mx 1 b
}
v
1.34
v 2
5*
1}
1.34 2
} 5 Ï *F
y 5 mx 1 b
x 5 my 1 b
x 2 b 5 my
A water line length of about 31.3 feet is needed to
achieve a maximum speed of 7.5 knots.
b
1
}x 2 } 5 y
m
m
A 5 0.2195h0.3964
50.
b
1
(x) 5 }
x2}
m
m
21
A
0.2195
} 5 h0.3964
b
1
Slope: }
m
y-intercept: 2}
m
This is the graph of a line, and it is a function if m Þ 0.
Problem Solving
46.
A
1}
0.2195 2
1/0.3964
A
1}
0.2195 2
2.523
5 (h0.3964)1/0.3964
5h
When A 5 1.6: h 5 1 }
0.2195 2
1.6
E 5 0.81419D
E
0.81419
}5D
E
250
5}
ø 307
When E 5 250: D 5 }
0.81419
0.81419
51. a.
1
g(x) 5 2x
x
1
* 2 3 5 0.5w
*23
0.5
}5w
2* 2 6 5 w
g(x) 5 2x
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
b. When * 5 6.5: w 5 2(6.5) 2 6 5 7
y 5 2x
The weight is 7 pounds.
x 5 2y
5
C 5 }9 (F 2 32)
9
5
2x 5 y
2x 5 g21(x)
} C 5 F 2 32
9
5
}C 1 32 5 F
b.
y
g(x) 5 2x 1 7
4
From the inverse, you can convert temperatures from
degrees Celsius C to degrees Fahrenheit F.
4
9
b. When C 5 5: F 5 } (5) 1 32 5 9 1 32 5 41
5
At the end of the race, the temperature was 148F.
c.
x
g(x) 5 2x 2 5
At the start of the race, the temperature was 418F.
9
When C 5 210: F 5 }5 (210) 1 32 5 218 1 32 5 14
ø (7.29)2.523 ø 150
The function g(x) 5 2x is
its own inverse because the
graph of an inverse relation is
a reflection of the graph of the
original relation in the line
y 5 x and the graph of an
inverse relation is the same
as the graph of the original
relation.
y
* 5 0.5w 1 3
48. a.
2.523
The height of a 60 kilogram person who has a
body surface area of 1.6 square meters is about 150
centimeters.
The amount that could be obtained for 250 euros was
about $307.
47. a.
7.5 2
When v 5 7.5: * 5 1 }
ø 31.3
1.34 2
x2b
}5y
m
f
}
v 5 1.34Ï*F
49.
g(x) 5 2x 1
1
2
c. g(x) 5 2x 1 b, where b is any real number.
Mixed Review for TAKS
52. A;
f(x) 5 25x 4 1 3x 3 1 10x 2 2 x 2 8
f(21) 5 25(21)4 1 3(21)3 1 10(21)2 2 (21) 2 8
Intersection
X=-40
Y=-40
The temperature that is the same on both temperature
scales is 240.
f(21) 5 25 2 3 1 10 1 1 2 8
f (21) 5 25
53. G;
s 1 a 5 750
s 5 30 1 3a
Algebra 2
Worked-Out Solution Key
339
Chapter 6,
continued
Quiz 6.3–6.4 (p. 445)
3
11.
1. f (x) 1 g(x) 5 4x 2 2 x 1 2x 2 5 6x 2 2 x
2. g (x) 2 f (x) 5 2x 2 2 (4x 2 2 x)
3
f (g (x)) 5 f
56
5. f (g(x)) 5 f (2x2) 5 4(2x 2)2 2 2x 2 5 4(4x 4) 2 2x2
f
(x) 5 23x 1 15
f(x) 5 x 2 2 16, x q 0
y 5 x 2 2 16
2
x 5 y 2 2 16
x 1 16 5 y 2
}
6Ï x 1 16 5 y
}
f 21(x) 5 Ïx 1 16
The domain of g (g (x)) consists of all real numbers.
3
3
g ( f (x)) 5 g( x 2 9)
2
y 5 2}9x 5
5x2919
2
5x
x 5 2}9 y 5
9
2}2x 5 y5
g (f ( x)) 5 g (5x 3)
Î
Î29x2 5 y
}
5
}
5
3
5x
5
3
}
5 51 }5 2
5 Îx 3
5x
5x
x
2
f(x) 5 2}9x 5
15.
f (x) 5 x 2 9, g(x) 5 x 1 9
1Î}5x 2
3 }
}
Î29x2 ++1616 5 y
}
5
}
5}
Ï2144x
2
}5y
5}
Ï2144x
f 21(x) 5 }
2
340
Algebra 2
Worked-Out Solution Key
2 1/2
5x
21
14.
The domain of f ( f (x)) consists of all real numbers.
55
1 6x 2
5x
23x 1 15 5 y
8. g(g(x)) 5 g(2x 2) 5 2(2x2)2 5 2(4x 4) 5 8x 4
}
2
1/2
1
5 64x 2 32x 1 4x 2 4x 1 x
1Î 2
1 1 2 1 1/2
x 2 5 5 2}3 y
5 64x4 2 32x 3 1 x
f ( g (x)) 5 f
2
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
2
5 4(16x4 2 8x 3 1 x2) 2 4x 2 1 x
3 x
}
5
1 6x
5 }
6
1
5 4(4x 2 x)(4x 2 x) 2 4x 1 x
}
11
x 5 2}3 y 1 5
The domain of g( f (x)) consist of all real numbers.
Î
2
1
7. f ( f (x)) 5 f (4x 2 2 x) 5 4(4x 2 2 x)2 2 (4x2 2 x)
10.
x 2 1 1/2
6
}
g ( f (x)) 5 g (6x 2 1 1)
y 5 2}3 x 1 5
5 2(16x 4 2 8x 3 1 x 2) 5 32x 4 2 16x 3 1 2x 2
x
f (x) 5 5x 3, g(x) 5 3 }5
G
2 G
2
x 2 1 1/2
6
}
2
5 (x2)
5 2(4x 2 x)(4x 2 x)
}
F1
F1
1
5x2111
2
5x
2
1
5 }
6
6. g( f (x)) 5 g(4x 2 2 x) 5 2(4x 2 2 x)2
5x1929
1
1
f (x) 5 2}3 x 1 5
13.
f ( g (x)) 5 f (x 1 9)
1
5 61}
11
6 2
The domain of f (g(x)) consists of all real numbers.
9.
1
x21
to all real numbers except x 5 0.
2
3
2
5x
f
3
5 2}3 1 2 }2 x 1 }4
1
5x
numbers. Because g(0) 5 0, the domain of }g is restricted
4
1
5 x 2 }6 1 }6
The function f and g each have the same domain: All real
2
2
1
5 x 2 }4 1 }4
f (x)
4x 2 2 x
4x 2
x
1
4. } 5 }
5 }2 2 }2 5 2 2 }
2
2x
g(x)
2x
2x
2x
2
3
x 2 1 1/2
12. f (x) 5 6x 2 1 1, x q 0; g (x) 5 }
6
The function f and g each have the same domain:
All real numbers. So, the domain of f + g also consists of
all real numbers.
5 16x 4 2 2x 2
g (f (x)) 5 g 1 2}2 x 1 }4 2
1
1
3. f (x) + g(x) 5 (4x 2 2 x)(2x 2) 5 8x 4 2 2x 3
1
5 2}21 2}3x 1 }6 2 1 }4
1
1 }6
5 2x 2 2 4x 2 1 x 5 22x 2 1 x
The function f and g each have the same domain: All real
numbers. So, the domain of f 2 g also consists of all real
numbers.
2
f (g (x)) 5 f 1 2}3x 1 }6 2
2
The function f and g each have the same domain: All real
numbers. So, the domain of f 1 g also consists of all real
numbers.
2
1
f (x) 5 2}2 x 1 }4, g (x) 5 2}3 x1 }6
Chapter 6,
16.
continued
f (x) 5 5x 1 12
y 5 5x 1 12
x 5 5y 1 12
x 2 12 5 5y
1
5
x
0
y
0
1
2
3
4
23 24.24 25.2 26
The domain is x q 0 and the range is y a 0.
12
5
}x 2 } 5 y
1 }
2. f (x) 5 }Ï x
4
1
12
f 21(x) 5 }5x 2 }
5
1
y
3
17.
f (x) 5 23x 2 4
y 5 23x 3 2 4
x 5 23y 3 2 4
x
21
3
x 1 4 5 23y
x14
23
} 5 y3
Î
}
3
Î
x14
23
}5y
0
f (x)
0 0.25 0.35 0.43 0.5
1
2
3
4
The domain is x q 0 and range is y q 0.
}
3
x
(x 1 4)9
23 + 9
}5y
13}
3. y 5 2}Ï x
2
3}
Ï9x 1 36
5y
2}
3
y
3}
Ï9x 1 36
1
f 21(x) 5 2}
3
x
22
f (x) 5 9x 4 2 49, x a 0
18.
y 5 9x 4 2 49
x 5 9y 4 2 49
x
x 1 49 5 9y 4
f (x)
x 1 49
9
22
21
0
1
2
20.63
0.5
0 20.5 20.63
} 5 y4
Î
The domain and range are all real numbers.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
}
4
x 1 49
9
3}
4. g (x) 5 4Ï x
6 }5y
Ï
4
}
y
(x 1 49)9
5y
6 }
9+9
4}
Ï9x 1 441
5y
6}
3
2
4}
f
21
x
22
Ï9x 1 441
3
(x) 5 2}
19. C(g (d)) 5 C(0.02d) 5 2.15(0.02d) 5 0.043d
C(g (400)) 5 0.043(400) 5 17.2
x
The expression C(g(400)) or $17.20 represents the cost
of gasoline for the car that is driven 400 miles.
f (x)
Lesson 6.5
22
21
25.04 24
0 1
2
0 4 5.04
The domain and range are all real numbers.
6.5 Guided Practice (pp. 447 – 449)
5.
}
1. y 5 23Ï x
1
21
y
x
X=.80851064
Y=.99808114
A pendulum with a period of 1 second is about
0.81 foot long.
Algebra 2
Worked-Out Solution Key
341
P. 449-451
Chapter 6,
continued
11. g(x) 5 2Î x 1 2 2 3
}
3 }
6. y 5 24Ï x 1 2
y
1
1
21
y
x
21
x
}
Because h 5 0 and k 5 2, shift the graph of y 5 24Ï x
up 2 units.
The domain is xq0 and the range is ya2.
}
7. y 5 2Ï x 1 1
Because h 5
22 and k 5 23, shift the graph of
3}
g(x) 5 2Ï x left 2 units and down 3 units.
The domain and range are both all real numbers.
6.5 Exercises (pp. 449–451)
y
Skill Practice
1. Square root functions and cube root functions are
examples of radical functions.
}
2. a. When a 5 23, the graph of y 5 Ï x is stretched
1
vertically by a factor of 3 and then reflected in the
x-axis.
x
21
}
Because h 5 1 and k 5 0, shift the graph of y 5 2Ï x left
1 unit. The domain is xq21and the range is yq0.
1 }
8. f (x) 5 }Ï x 2 3 2 1
2
}
b. When h 5 2, the graph of y 5 Ï x is shifted to the
right 2 units.
}
c. When k 5 4, the graph of y 5 Ï x is shifted up 4 units.
}
3. y 5 24Ï x
y
1
x
x
0
y
0 24 25.66 26.93 28
2
3
4
y
Because h 5 3 and k 5 21, shift the graph of
1
1
1
21
}
f (x) 5 }2 Ï x right 3 units and down 1 unit.
x
The domain is xq3 and the range is f (x)q21.
3}
9. y 5 2Ï x 2 4
y
The domain is xq0 and the range is ya0.
1
1 }
4. f (x) 5 }Ï x
2
x
21
3}
Because h = 4 and k 5 0, shift the graph of y 5 2Ï x
right 4 units.
x
0
f (x)
0 0.5 0.71 0.87 1
1
2
3
4
y
The domain and the range are both all real numbers.
10. y 5 Î x 2 5
3 }
1
1
y
21
x
21
x
The domain is x q 0 and the range is f (x)q0.
4 }
5. y 5 2}Ï x
5
3}
Because h 5 0 and k 5 25, shift the graph of y Ï x down
5 units.
The domain and range are both all real numbers.
342
Algebra 2
Worked-Out Solution Key
x
0
y
0 20.8 21.13 21.39 21.6
1
2
3
4
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
21
Chapter 6,
continued
13}
10. y 5 }Î x
4
y
1
x
21
x
y
22
0
21
1
2
20.32 20.25 0 0.25 0.31
The domain is xq0 and the range is ya0.
y
}
6. y 5 26Ï x
1
4
x
0
y
0 26 28.49 210.39
1
2
3
x
22
4
212
The domain and range are all real numbers.
y
2
11. y 5 2Î x
3 }
x
21
x
y
21 0 1
22
22.52 22 0 2
2
2.52
y
The domain is xq0 and the range is ya0.
1
}
7. y 5 5Ï x
x
22
x
0 1
y
0 5 7.07 8.66 10
2
3
4
The domain and range are all real numbers.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
y
12. f (x) 5 25Î x
3 }
x
22 21 0
f (x)
6.3
5
2
1
2
0 25 26.3
y
x
21
The domain is xq0 and the range is yq0.
2
}
8. g(x) 5 9Ï x
x
22
x
0
1
g(x)
0
9
2
3
4
12.73 15.59 18
The domain and range are all real numbers.
y
13}
13. h(x) 5 2}Î
x
7
3
x
22
h(x)
0.18 0.14 0 20.14 20.18
21
0
1
2
x
21
The domain is xqand the range is g(x)q0.
9. D;
3
y
0.3
}
y 5 2}2Ï x
22
x
0
y
0 21.5 22.12 22.6 23
1
2
3
x
4
The domain and range are all real numbers.
Algebra 2
Worked-Out Solution Key
343
Chapter 6,
continued
14. g(x) 5 6Î x
}
3 }
x
g(x)
18. y 5 24Ï x 2 5 1 1
22
21
0 1
2
27.56
26
0 6 7.56
y
1
x
21
y
3
}
Because h 5 5 and k 5 1, shift the graph of y 5 24Ï x
right 5 units and up 1 unit.
x
22
The domain is xq5 and the range is ya1.
3
33}
19. y 5 }x1/321 5 }Î x 21
4
4
The domain and range are all real numbers.
73}
15. y 5 }Î x
9
x
1
x
22
22
h(x)
y
0
21
1
2
20.98 20.78 0 0.78 0.98
Because h 5 0 and k 5 21, shift the graph of
3
y
y 5 }4x1/3 down 1 unit.
1
22
The domain and range are both all real numbers.
x
3
20. y 5 22Î
x1515
}
y
The domain and range are all real numbers.
}
16. f (x) 5 2Ï x 2 1 1 3
1
x
22
Because h 5 25 and k 5 5, shift the graph of
3}
y 5 22Ï x left 5 units and up 5 units.
1
The domain and range are both all real numbers.
x
21
}
Because h 5 1 and k 5 3, shift the graph of f (x) 5 2Ï x
right 1 unit and up 3 units.
The domain is xq1 and the range is f (x)q3.
}
17. y 5 (x 1 1)1/2 1 8 5 Ï x 1 1 1 8
21. h(x) 5 23Î x 1 7 2 6
3 }
2
22
y
x
y
2
21
x
Because h 5 21 and k 5 8, shift the graph of y 5 x1/2
left 1 unit and up 8 units.
The domain is xq21 and the range is yq8.
344
Algebra 2
Worked-Out Solution Key
Because h 5}27 and k 5 26, shift the graph of
3
h(x) 5 23Ï x left 7 units and down 6 units.
The domain and range are both all real numbers.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
y
Chapter 6,
continued
}
}
22. y 5 2Ï x 2 4 2 7
2
29. The function y 5 Ï x 2 12 is a translation of the function
}
y 5 Ï x that has a domain xq0 and a range of yq0.
Because h 5 12, the domain is xq12. Because k 5 0,
the range is yq0.
y
x
22
1 }
30. The function y 5 }Ï x 2 4 is a translation of the function
3
1 }
y 5 }3Ï x that has a domain of xq0 and a range of yq0.
Because h 5 0, the domain is xq0. Because k 5 24, the
range is yq24.
}
Because h 5 4 and k 5 27, shift the graph of y 5 2Ï x
right 4 units and down 7 units.
The domain is xq4 and the range is ya27.
13}
23. g(x) 5 2}Ï x 2 6
3
1
domain and range of all real numbers.
3
32. The function g(x) 5 Î
x 1 7 is a translation of the
y
22
13}
31. The function y 5 }Î x 1 7 is a translation of the
2
13}
function y 5 }2Ï x that has a domain and range of all real
13}
numbers. Therefore, the function y 5 }2Îx 1 7 also has a
}
function g(x) 5 Î x that has a domain of all real
3 }
numbers. Therefore, the function g(x) 5 Î
x 1 7 also has
a domain and range of all real numbers.
3 }
x
1
}
33. The function f (x) 5 }4 Ïx 2 3 1 6 is a translation of the
1
}
function f (x) 5 }4 Ï x that has a domain of xq0 and a
range
of f (x)q0. Because h 5 3, the domain is xq3. Because
k 5 6, the range is f (x)q6.
Because h 5 0 and k 5 26 shift the graph of
1
3}
g(x) 5 }3Ï x down 6 units.
34.
The domain and range are both all real numbers.
24. y 5 4Î x 2 4 1 5
3 }
y
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
n}
When n is even, the domain of the function y 5 Ï x is
xq0 and the range of the function is yq0. When n is}
n
odd, the domain and the range of the function y 5 Ï x are
both all real numbers.
2
x
21
3}
Because h 5 4 and k 5 5, shift the graph of y 5 4Ï x
right 4 units and up 5 units.
Problem Solving
The domain and range are both all real numbers.
35.
}
25. The domain of y 5 Ï x 2 5 1 4 is limited because of the
square root of a negative number is not a real number.
Because the domain is restricted, the range is also
affected.
X=43.03
Y=8
26. The error occured in describing the horizontal
translation. Because h 5 21, the graph is translated left
3 }
1 unit. So, graph of y 5 22Îx 1 1 2 3 is the graph of
3 }
y 5 22Î x translated left 1 unit and down 3 units.
You can see 8 miles at an altitude of about 43 feet above
sea level.
36. a.
27. C;
Because the graph of y 5 3Î x is shifted left 2 units,
h 5 22 and k 5 0. An equation of the translated graph is
3 }
y 5 3Îx 1 2 .
3 }
X=2
.01
Y=1
.5
7
}
28. The function y 5 Ï x 1 5 is a translation of the function
}
y 5 Ï x that has a domain of xq0 and a range of yq0.
Because h 5 25, the domain is xq25. Because k 5 0,
the range is yq0.
A pendulum with a length of 2 feet has a period of
about 1.6 seconds.
Algebra 2
Worked-Out Solution Key
345
Chapter 6,
continued
b.
s 5 :r 1 :r 2
40. a.
s 5 :(r 1 r 2)
s 1 :1 }4 2 5 :1 r 2 1 r 1 }4 2
1
X=3.26
1
:
Y=2
s1}
5 :1 r 1 }2 2
4
A pendulum with a period of 2 seconds is about
3.3 feet long.
1 2
:
42
1
:1
1 2
2
1
}F
s1} 5 r1}
Î
273.15 1 C
5 331.5Î}
273.15
C
5 331.5Î1 1 }
273.15
}
}
:
2
Ï}:F1 s 1 }4 2 5 r 1 }2
K
37. a. v 5 331.5 } , Kq0
273.15
1
}
1
Ï:
}
:
Ï1 s 1 }4 2 5 r 1 }2
}
}
}
1
Ï:
1
:
4
}
Ï
1
1
2
}
} s 1 } 2 } 5 r
b. Because Kq0 and K 5 273.15 1 C:
b.
273.15 1 Cq0
Cq2273.15
The domain of the function is Cq2273.15.
The range of the function is vq0.
s
250
200
150
100
50
:
1
1
c. r 5 }
} s 1 } 2 }
4
2
Ï :F
}
Ï
1
0
0
1000 2000 3000 p
Power (horsepower)
3:
1
Ï :F
Î
Î
}
}
W
165
39. a. vt 5 33.7 } 5 33.7 }
A
A
A
2
vt
c.
Ï
:
1
4
}
1
r5}
2 }2
} + Ï :F
The power of a 3500 pound car that reaches a speed of
200 miles per hour is about 2470 horsepower.
b.
}
r5}
} } 1 } 2 }
4
2
Ï :F 4
r 5 0.5
The radius is 0.5 unit.
Mixed Review for TAKS
6
8
10
306.1 216.44 176.72 153.05 136.89
41. B;
y 5 ax 2 1 bx 1 c
0 5 a(0)2 1 b(0) 1 c
Vt
500
05c
37 5 a(1)2 1 b(1) 1 c
400
37 5 a 1 b 1 c
300
58 5 a(2)2 1 b(2) 1 c
200
58 5 4a 1 2b 1 c
100
Substitute c 5 0 into the last two equations.
0
0
2
4
6
8
10 A
a 1 b 1 c 5 37
4a 1 2b 1 c 5 58
a 1 b 1 0 5 37
4a 1 2b 1 0 5 58
a 1 b 5 37
4a 1 2b 5 58
a 1 b 5 37
4a 1 2b 5 58
346
Algebra 2
Worked-Out Solution Key
3 22
22a 2 2b 5 274
4a 1 2b 5
58
2a
5 216
a
5 28
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Speed (mph)
38.
Chapter 6,
continued
5. 3x3/2 5 375
Substitute a 5 28.
x3/2 5 125
a 1 b 5 37
(x3/2)2/3 5 1252/3
28 1 b 5 37
x 5 25
b 5 45
Check: 3(25)3/2 0 375
So a 5 28, b 5 45, and c 5 0.
2
3(125) 0 375
y 5 ax 1 bx 1 c
2
375 5 375 y 5 28x 1 45x 1 0
y 5 45x 2 8x 2
6. 22x3/4 5 216
x3/4 5 8
42. H;
(x )
3/4 4/3
(1128 1 628 1 908) 1 x8 5 3608
264 1 x 5 360
5 84/3
x 5 16
Check: 22(16) 3/4 0 216
22(8) 0 216
x 5 96
(y 2 72)8 5 968
216 5 216 y 5 168
2
7. 2}x1/5 5 22
3
Lesson 6.6
x1/5 5 3
6.6 Guided Practice (pp. 452 – 455)
(x )
1/5 5
3
1. Î x 2 9 5 21
}
5 35
x 5 243
Î3 }x 5 8
}
(Î3 x )3 5 83
2
Check:2}3(243)1/5 0 22
2
x 5 512
Check: Î512 2 9 0 21
8 2 9 0 21
2}3(3) 0 22
3 }
21 5 21 2.
22 5 22 8.
[(x 1 3)5/2]2/5 5 322/5
}
Ïx 1 25 5 4
x1354
(Ïx 1 25 )2 5 42
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
}
x51
x 1 25 5 16
Check: (1 1 3)5/2 0 32
45/2 0 32
x 5 29
}
Check: Ï29 1 25 0 4
}
Ï16 0 4
32 5 32 454
3.
2Îx 2 3 5 4
3 }
}
3
(x 2 5)
5 81
[(x 2 5)4/3]3/4 5 813/4
x 2 5 5 6(3)3
5 23
x 2 5 5 627
x2358
x 5 32
x 5 11
or
Check x 5 32:
3 } 0
11 2 3 4
Check: 2Î
3 } 0
8 4
2Î
454
4.
9.
4/3
x 2 5 5 6(811/4)3
Î3 }
x2352
(Ï3 x 2 3 )
(x 1 3)5/2 5 32
}
v(p) 5 6.3Ï1013 2 p
Check x 5 222:
(32 2 5)4/3 0 81
274/3 0 81
(222 2 5)4/3 0 81
(227)4/3 0 81
(23)4 0 81
81 5 81 81 5 81 }
48.3 5 6.3Ï1013 2 p
}
7.67 ø Ï 1013 2 p
}
2
(7.67)2 ø (Ï1013 2 p )
58.8 ø 1013 2 p
2954.2 ø 2p
954.2 ø p
The air pressure at the center of the hurricane is about
954 millibars.
x 5 222
10. (x 1 2)2/3 1 3 5 7
(x 1 2)2/3 5 4
[(x 1 2)2/3]3/2 5 43/2
x 1 2 5 (41/2)3
x 1 2 5 (62)3
x 1 2 5 68
x56
or
x 5 210
Algebra 2
Worked-Out Solution Key
347