Math 112 Chapter 4 Practice Test
Name___________________________________
Find the inverse of the relation.
8) y = 5 x
1) {(2, -14), (-5, -14), (-19, 12)}
y
10
2) {(-2, -7), (2, 7), (-1, 9), (1, -9)}
5
Find an equation of the inverse relation.
3) y = 5x - 2
-10
-5
5
10
x
-5
4) x3 y = 6
-10
5) x = y - 6y2
Provide an appropriate response.
Graph the equation as a solid line. Graph the inverse
relation as a dashed line on the same axes by reflecting
across the line y = x.
4
6) f(x) = x + 4
3
9) Prove that the function f is one-to-one.
1
f(x) = 6 - x
2
y
10) Prove that the function f is one-to-one.
3
f(x) = x + 5
10
5
-10
-5
5
10
11) Prove that the function f is not one-to-one.
f(x) = 2 - x4
x
-5
Using the horizontal-line test, determine whether the
function is one-to-one.
-10
12)
y
10
7) x = 4 y
5
y
10
-10
5
-10
-5
-5
5
-5
5
10
-10
x
-5
-10
1
10
x
13) f(x) = x3 - 3x + 3
8
21) f(x) = 3x - 9
8x - 6
y
3
22) f(x) = x - 7
4
Graph the inverse of the function plotted, on the same set
of axes. Use a dashed curve for the inverse.
-8
-4
8 x
4
23)
y
-4
10
-8
5
3
14) f(x) = x + 5 - 5
-10
-5
5
10
x
10
x
-5
8
y
-10
4
24)
-12
-8
-4
4
8
12
x
y
10
-4
-8
-10
Determine whether the function is one-to-one by graphing
and using the horizontal line test.
x - 10
15) f(x) = x + 3
-10
16) f(x) = -x3 + 7
For the function f, use composition of functions to show
that f -1 is as given.
Find the inverse of the function.
25) Let f(x) = 17) f(x) = x + 4
x + 9
. Show that f-1 (x) = 7x - 9.
7
1
18) f(x) = x + 7
5
7
26) Let f(x) = (7 + x)/x. Show that f-1 (x) = .
x - 1
3
19) f(x) = x - 3
3
5
27) Let f(x) = x. Show that f-1 (x) = x.
5
3
Determine whether the given function is one -to-one. If it is
one-to-one, find a formula for the inverse.
Find the domain and range of the inverse of the given
function.
20) f(x) = 7x3 + 8
28) f(x) = x3 - 1
2
36) f(x) = 4 - e-x
29) f(x) = x2 - 8; x ≥ 0
30) f(x) = y
x + 1
x - 3
10
5
Graph the one-to-one function as a solid curve and its
inverse as a dashed curve on the same axes.
5
31) f(x) = x + 2
3
-10
-5
5
10
x
-5
y
10
-10
5
A)
-10
-5
5
10
y
x
10
-5
5
-10
-10
-5
32) y = x3 + 3
5
10
x
5
10
x
5
10
x
-5
y
-10
10
B)
y
10
-10
10
x
5
-10
-10
-5
-5
Evaluate to four decimal places using a calculator.
33) e 4.29
-10
C)
34) e2.041
y
10
35) e3 2
5
Choose the graph that matches the function.
-10
-5
-5
-10
3
D)
C)
y
y
-10
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
5
10
x
5
10
x
D)
37) f(x) = 2 x + 3
y
10
y
10
5
5
-10
-10
-5
5
10
-5
x
-5
-5
-10
-10
38) f(x) = 4 x+1 - 3
A)
y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
5
-5
-5
-10
-10
B)
10
A)
y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
5
-5
-5
-10
-10
4
10
x
x
Graph the function.
B)
39) f(x) = - 4 x
y
10
y
5
5
4
3
-10
-5
5
10
2
x
1
-5
-5 -4 -3 -2 -1
-1
1
2
3
4
5
x
-2
-10
-3
-4
C)
-5
y
10
40) f(x) = 5 (x - 4)
5
6
y
4
-10
-5
5
10
x
2
-5
-6
-4
-2
-10
2
6 x
4
-2
-4
D)
y
-6
10
41) f(x) = 2 - x - 3
5
y
-10
-5
5
10
x
5
4
-5
3
2
1
-10
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
5
1
2
3
4
5
x
42) f(x) = e3x - 3
45) f(x) = 4 x - 1; relative to f(x) = 4 x
y
y
5
10
4
3
5
2
1
-5 -4 -3 -2 -1
-1
1
2
3
4
x
5
-10
-5
-2
5
x
10
-5
-3
-4
-10
-5
46) f(x) = 2 - 1 -x; relative to f(x) = 1 x
43) f(x) = ex - 8
y
y
10
10
5
5
-5 -4 -3 -2 -1
1
2
3
4
5x
-10
-5
5
-5
-5
-10
-10
Graph the function. Describe its position relative to the
graph of the indicated basic function.
x
10
47) f(x) = e 3x - 2; relative to f(x) = e x
y
44) f(x) = -4 x+2 ; relative to f(x) = 4 x
10
y
10
5
5
-10
-10
-5
5
10
-5
5
-5
x
-5
-10
-10
6
10
x
48) f(x) = e-0.3x; relative to f(x) = ex
10
55) Suppose the amount of a radioactive element
remaining in a sample of 100 milligrams after x
years can be described by A(x) = 100e-0.0137x.
y
How much is remaining after 168 years? Round
the answer to the nearest hundredth of a
milligram.
5
-10
-5
10 x
5
56) The number of bacteria growing in an
incubation culture increases with time according
to B = 7100(2)x, where x is time in days. Find the
-5
number of bacteria when x = 0 and x = 5.
-10
Graph the function.
1y
57) x = 6
49) f(x) = e-x + 2 ; relative to f(x) = ex
y
20
5
y
10
-10
-5
5
10
x
-20
-10
10
20
x
-10
-20
-5
Find the amount that will be in an account, given the stated
conditions.
58) f(x) = ln x
50) P = $1000, t = 4, r = 9% compounded
semiannually
6
y
4
51) P = $480, t = 2, r = 14% compounded quarterly
2
-6
52) P = $800, t = 11, r = 8% compounded
continuously
-4
-2
2
-2
-4
Solve the problem.
-6
53) An initial investment of $1000 is appreciated for
2 years in an account that earns 6% interest,
compounded semiannually. Find the amount of
money in the account at the end of the period.
Find the value of the expression.
59) log 4
4
54) An initial investment of $700 is appreciated for 2
years in an account that earns 9% interest,
compounded continuously. Find the amount of
money in the account at the end of the period.
60) log
10
1000
61) log18 18
7
4
6 x
Graph the function and its inverse using the same set of
axes. Use any method.
62) log5 5 2
80) f(x) = log5 x; f-1 (x) = 5 x
63) log5 5 7/4
5
Convert to a logarithmic equation.
y
64) 161/2 = 4
65) e-3 = t
5 x
-5
66) yz = 6
67) e-t = 25
-5
Convert to an exponential equation.
81) f(x) = ex; f-1 (x) = ln x
68) log8 512 = t
5
y
69) ln 33 = 3.4965
70) log2 T8 = 8
5 x
-5
71) log
w
Q = 12
72) log10 16 = 1.2041
-5
Find the following using a calculator. Round to four
decimal places.
Graph the function. Describe its position relative to the
graph of the indicated basic function.
73) log 0.46
82) f(x) = log3 (x - 5); relative to f(x) = log3 x
y
74) ln 267
10
75) ln 0
5
Find the logarithm using the change-of-base formula.
-10
76) log100 20
-5
5
-5
77) log100 20
-10
78) log7.4 4.6
79) log5.9 1600
8
10
x
92) logx 2yz
83) f(x) = ln(x + 3); relative to f(x) = ln x
y
10
Express as a product.
93) log 10 x10
5
-10
-5
5
10
94) ln y45
x
95) log b M-8
-5
-10
96) ln 84) f(x) = 6 - log2 x; relative to f(x) = log2 x
y
5
-5
98) log
5
8
Express as a difference of logarithms.
x
97) ln 7
10
-10
3
10
M
g 28
x
99) loga -5
E
L
Express in terms of sums and differences of logarithms.
-10
100) loga 4x3 yz2
Find the domain and the vertical asymptote of the
101) log 85) f(x) = ln (6 - x)
86) f(x) = 5 + log4 x
x5 z
y2
m 3 p4
102) logb
n 5 b9
87) f(x) = ln x - 3
Solve.
103) logb
x4 y2
z6
104) logb
4 x5 b2
y4 z 12
88) If an earthquake measured 8.5 on the Richter
scale, what was the approximate intensity of the
earthquake in terms of I0 ? The magnitude on
the Richter scale of an earthquake of intensity I
is log10(I/Io).
Express as a single logarithm and, if possible, simplify.
Express as a sum of logarithms.
105) loga 0.1 + loga 1000
89) log xy
6
106) ln 24 - ln 3
90) log (16 · 8)
2
107) 5 log a q - log a r
6
91) log4 Z
5
9
108)
1
loga x + 4 loga y - 3 loga x
2
Simplify.
124) ln e4
109) log(t3 + 729) - log(t + 9)
125) 3 log3 (6x)
110) logb x5 - 2logb x
126) log e e x - 10
111) log a
4
- log a 4x
x
127) log 10-19
Solve the exponential equation.
2
112) [ln (k2 - 144) - ln(k + 12)] + ln (k + y)
3
128) 3 (12 - 3x) = 729
129) 4 (1 + 2x) = 1024
5
113) 6 ln x3 - 5 ln y4
130) 5 7x = 5
114) log(t3 + 125) - log(t + 5)
1
131) 3 x2 + 5 x = 81
115) logb x7 - 2logb x
116) log a
132) 22x = 10-4x
3
- log a 3x
x
133) e-x = 4 2x
2
117) [ln (z 2 - 100) - ln(z + 10)] + ln (z + y)
3
134) e-0.05t = 0.18
5
118) 8 ln x3 - 10 ln y4
135) 300 - (1.75)x = 0
Solve.
136) ex - 5e-x = 4
119) Given that loga 11 = 2.398, and loga 5 = 1.609,
1
find loga .
55
137)
120) Given that loga 5 = 0.699, and loga 2 = 0.301, find
loga 125.
8 x - 8 -x
= 5
8 x + 8 -x
Solve the logarithmic equation.
1
138) log x = 25
2
121) Given that loga 2 = 0.3010 and loga 3 = 0.4771,
find loga 18.
139) log x = 4
140) ln(3x - 4) = ln 4 - ln (x - 1)
122) Let log b A = 1.988 and log b B = 0.155. Find
log b AB.
141) log 5 x = log 4 + log (x + 4 )
123) Given log b 3 = 0.8397 and log b 7 = 1.4873,
evaluate log b 3b .
142) log5 (8x - 7) = 2
10
143) log (x + 10) - log (x + 4) = log x
11
Answer Key
Testname: MATH 112 CHAPTER 4 PRACTICE TEST
1) {(-14, 2), (-14, -5), (12, -19)}
2) {(-7, -2), (7, 2), (9, -1), (-9, 1)}
3) x = 5y - 2
4) y3 x = 6
5) y = x - 6x2
6)
y
10
5
-10
-5
5
10
x
5
10
x
5
10
x
-5
-10
7)
y
10
5
-10
-5
-5
-10
8)
y
10
5
-10
-5
-5
-10
12
Answer Key
Testname: MATH 112 CHAPTER 4 PRACTICE TEST
9) Assume that f(a) = f(b) for any numbers a and b in the domain of f.
1
1
Since f(a) = 6 - a and f(b) = 6 - b, we have
2
2
1
1
6 - a = 6 - b
2
2
1
1
- a = - b
2
2
a = b
Thus if f(a) = f(b) then a = b and f is one-to-one.
10) Assume that f(a) = f(b) for any numbers a and b in the domain of f.
3
3
Since f(a) = a + 5 and f(b) = b + 5, we have
3
3
a + 5 = b + 5
3
3
a = b
a = b
Thus if f(a) = f(b) then a = b and f is one-to-one.
11) Answers may vary. Possible answer:
Find two numbers a and b for which a ≠ b and f(a) = f(b). Two such numbers are -2 and 2 because f(-2) = f(2) = -14.
Thus, f is not one-to-one.
12) Yes
13) No
14) Yes
15) Yes
16) Yes
17) f-1 (x) = x - 4
18) f-1 (x) = 5x - 35
19) f-1 (x) = (x + 3)3
20) f-1 (x) = 3 x - 8
7
6x - 9
21) f-1 (x) = 8x - 3
22) f-1 (x) = x3 + 7
23)
y
10
5
-10
-5
5
10
x
-5
-10
13
Answer Key
Testname: MATH 112 CHAPTER 4 PRACTICE TEST
24)
y
10
-10
10
x
-10
25) Answers may vary. One possible solution is:
1. (f-1 ∘ f)(x) = f-1 (f(x)) = f-1 x + 9 /7 = 7 x + 9 /7 - 9 = x + 9 - 9 = x;
2. (f ∘ f-1 )(x) = f(f-1 (x)) = f 7x - 9 = 7x - 9 + 9 /7 = 7x/7 = x
26) Answers may vary. One possible solution is:
1. (f-1 ∘ f)(x) = f-1 (f(x)) = f-1 7 + x /x = 2. (f ∘ f-1 )(x) = f(f-1 (x)) = f
7
=
x - 1
7 + 7
7
7
= = = x;
7 + x
7 + x - x
7
- 1
x
x
x
7
x - 1
7
x - 1
= 7 x - 1 + 7
x - 1
7
x - 1
3 5
5
x = x;
27) 1. (f-1 ∘ f)(x) = f-1 (f(x)) = f-1 x = 5 3
3
5 3
3
x = x
2. (f ∘ f-1 )(x) = f(f-1 (x)) = f x = 3 5
5
28) Domain and range: all real numbers
29) Domain: [-8, ∞); range: [0, ∞)
30) Domain: (-∞, 1) ∪ (1, ∞)
Range: (-∞, 3) ∪ (3, ∞)
31)
y
10
5
-10
-5
5
10
x
-5
-10
14
= 7x
= x
7
Answer Key
Testname: MATH 112 CHAPTER 4 PRACTICE TEST
32)
y
10
-10
10
x
5
x
-10
33) 72.9665
34) 7.6983
35) 403.4288
36) D
37) C
38) B
39)
y
5
4
3
2
1
-5 -4 -3 -2 -1
-1
1
2
3
4
-2
-3
-4
-5
40)
6
y
4
2
-6
-4
-2
2
4
6 x
-2
-4
-6
15
Answer Key
Testname: MATH 112 CHAPTER 4 PRACTICE TEST
41)
y
5
4
3
2
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5
x
1
2
3
4
5
x
-2
-3
-4
-5
42)
y
5
4
3
2
1
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
43)
y
10
5
-5 -4 -3 -2 -1
1
2
3
4
5x
-5
-10
44) Moved left 2 unit(s);
reflected across x-axis
y
10
5
-10
-5
5
10
x
-5
-10
16
Answer Key
Testname: MATH 112 CHAPTER 4 PRACTICE TEST
45) Moved right 1 unit(s)
y
10
5
-10
-5
5
10
x
5
10
x
5
10
x
-5
-10
46) Reflected across y-axis;
reflected across x-axis;
moved up 2 unit(s)
y
10
5
-10
-5
-5
-10
47) Shrunk horizontally;
moved down 2 unit(s)
y
10
5
-10
-5
-5
-10
17
Answer Key
Testname: MATH 112 CHAPTER 4 PRACTICE TEST
48) Stretched horizontally;
reflected across the y-axis
y
10
5
-10
-5
10 x
5
-5
-10
49) Reflected across the y-axis;
moved right 2 units
y
20
10
-10
-5
5
10
x
10
20
x
-10
-20
50) $1422.10
51) $632.07
52) $1928.72
53) $1125.51
54) $838.05
55) 10.01 milligrams
56) 7100, 227,200
57)
5
-20
-10
y
-5
18
Answer Key
Testname: MATH 112 CHAPTER 4 PRACTICE TEST
58)
6
y
4
2
-6
-4
-2
2
4
6 x
-2
-4
-6
59) 1
60) 3
1
61)
2
62) 2
7
63)
4
64)
1
= log 4
16
2
65) ln t = -3
66) z = log y 6
67) ln 25 = -t
68) 8 t = 512
69) e3.4965 = 33
70) 2 8 = T8
71) w12 = Q
72) 101.2041 = 16
73) -0.3372
74) 5.5872
75) Does not exist
76) 0.6505
77) 0.6505
78) 0.7625
79) 4.1566
80)
5
y
5 x
-5
-5
19
Answer Key
Testname: MATH 112 CHAPTER 4 PRACTICE TEST
81)
5
y
5 x
-5
-5
82) Moved right 5 units
y
10
5
-10
-5
5
10
x
5
10
x
5
10
x
-5
-10
83) Moved left 3 units
y
10
5
-10
-5
-5
-10
84) Reflected across x-axis;
moved up 6 units
y
10
5
-10
-5
-5
-10
20
Answer Key
Testname: MATH 112 CHAPTER 4 PRACTICE TEST
85) Domain: (-∞, 6); vertical asymptote: x = 6
86) Domain: (0, ∞); vertical asymptote: x = 0
87) Domain: (0, ∞); vertical asymptote: x = 0
88) 320,000,000 · I0
89) log x + log y
6
6
90) log 16 + log 8
2
2
6
91) log4 + log4 Z
5
92) logx 2 + logx y + logx z
93) 10 log 10 x
94) 45 ln y
95) -8 log b M
96)
1
ln 8
3
97) ln x - ln 7
98) log M - log 28
g
g
99) loga E - loga L
100) loga 4 + 3loga x + loga y + 2loga z
101) 5 log x + log z - 2 log y
102) 3logbm + 4logbp - 5logbn - 9
103) 2logbx + logby - 3logbz
5
1
104) logbx + - logby - 3logbz
4
2
105) loga 100
106) ln(8)
q5
107) log a r
108) loga
y4
x5/2
109) log(t2 - 9t + 81)
110) logb x4
2
111) log a
x
112) ln (k - 12)2/3(k + y)
x9
113) ln y4
114) log(t2 - 5t + 25)
115) logb x6
3
116) log a
x
117) ln (z - 10)2/3(z + y)
x12
118) ln y8
119) -4.007
120) 2.097
121) 1.2552
21
Answer Key
Testname: MATH 112 CHAPTER 4 PRACTICE TEST
122)
123)
124)
125)
126)
127)
128)
129)
2.143
1.8397
4
6x
x - 10
-19
2
2
1
130)
7
-1 , -4
0
0
34.296
10.1923
1.6094
No solution
5
10,000
7
140)
3
131)
132)
133)
134)
135)
136)
137)
138)
139)
141) 16
142) 4
143) 2
22