MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Evaluate the expression using the values given in the table.
1) (g f)(1)
1)
x
1 5 11 12
f(x) -2 11 1 15
x
-5 -2 1 3
g(x) 1 -5 5 11
A) -2
B) 5
C) 11
D) -5
For the given functions f and g, find the requested composite function value.
t+ 3
;
Find (f g)(9).
2) f(t) = t4 + 18t2 + 81, g(t) =
3
A) 360
B) 625
C) 31
t+ 3
t4 + 24t2 + 144, g(t) =
;
3
3) f(t) =
A) 780
Find (f
2)
D) 25
g)(12).
B) 1369
3)
C) 37
D) 53
For the given functions f and g, find the requested composite function.
6
4
, g(x) =
;
Find (f g)(x).
4) f(x) =
x+1
3x
A)
5) f(x) =
4x + 4
18x
B)
x-8
, g(x) = 10x + 8;
10
A) x
18x
4 + 3x
Find (g
A) No, no
x-8
,
4
D)
18x
4 - 3x
5)
C) x -
g and g
4
5
D) x + 16
f, are equal to x.
B) No, yes
C) Yes, no
D) Yes, yes
g(x) = 4x + 8
A) No, no
8) f(x) =
6x
4 + 3x
f)(x).
B) 10x + 72
Decide whether the composite functions, f
6) f(x) = x , g(x) = x2
7) f(x) =
C)
4)
7)
B) No, yes
C) Yes, yes
D) Yes, no
1
, g(x) = x
x
A) No, no
6)
8)
B) Yes, yes
C) No, yes
1
D) Yes, no
For the given functions f and g, find the requested composite function.
5
5
, g(x) =
;
Find (f g)(x).
9) f(x) =
x+1
6x
10) f(x) =
x-2
, g(x) = 7x + 2;
7
Find functions f and g so that f
3
11) H(x) = x +1
Find (g
f)(x).
9)
10)
g = H.
11)
1
2
x -5
12)
13) H(x) = 4x + 8
13)
12) H(x) =
Find the domain of the composite function f
5
; g(x) = x + 6
14) f(x) =
x+9
15) f(x) = x + 6;
16) f(x) =
2
;
x+7
g(x) =
g.
14)
3
x+3
15)
-21
x
16)
g(x) =
17) f(x) =
x - 4; g(x) =
18) f(x) =
1
; g(x) =
x - 10
4
x-7
17)
x-1
18)
Find the inverse of the function and state its domain and range .
19) {(8, -11), (6, -10), (4, -9), (2, -8)}
19)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Indicate whether the function is one-to-one.
20) {(-12, 19), (-7, 19), (-14, 15)}
A) Yes
B) No
2
20)
Use the horizontal line test to determine whether the function is one-to-one.
21)
A) Yes
21)
B) No
SHOW YOUR WORK.
The graph of a one-to-one function f is given. Draw the graph of the inverse function f-1 as a dashed line or curve.
22) f(x) = x3 + 3
22)
Use the graph of the given one-to-one function to sketch the graph of the inverse function. For convenience, the graph of
y = x is also given.
23)
23)
3
Find the inverse function of f. State the domain and range of f.
3x - 2
24) f(x) =
x+5
24)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The function f is one-to-one. Find its inverse.
25) f(x) = x3 - 5
A) f-1 (x) =
26) f(x) =
3
x-5
B) f-1 (x) =
25)
3
C) f-1 (x) =
x+5
3
x-5
3
x+5
3
7x - 2
26)
-2x - 3
A) f-1 (x) =
7x
B) f-1 (x) =
7x - 2
C) f-1 (x) =
3
3 + 2y
D) f-1 (x) =
7y
27) f(x) =
D) f-1 (x) =
3 + 2x
7x
9
x-5
27)
-5 + 9x2
A) f-1 (x) =
x
B) f-1 (x) =
5x + 9
C) f-1 (x) =
x
-5 + 9x
D) f-1 (x) =
x
x
-5 + 9x
The graph of an exponential function is given. Match the graph to one of the following functions.
28)
A) f(x) = 3 x + 1
B) f(x) = 3 x
C) f(x) = 3 x + 1
4
D) f(x) = 3 x - 1
28)
Use transformations to graph the function. Determine the domain, range, and horizontal asymptote of the function.
29) f(x) = -2 x+3 + 4
29)
30) f(x) = 5 -x + 2
30)
Graph the function.
31) f(x) = 3 (x - 3) + 1
31)
5
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the equation.
1
32) 2 5 - 3x =
16
A)
32)
1
8
B) {3}
C) {-3}
D) {8}
1
33) 4 x =
64
33)
A) {-3}
B)
34) 2 x2 - 3= 64
A) {3, -3}
9+
6
C)
1
16
D) {3}
34)
B) {3}
35) 9 2x · 27(3 - x) =
A)
1
3
D) { 35, -
C) {6}
1
9
87 9 ,
35)
6
87
B) {-11}
C) {10}
D) {-8}
36) 8 x - 4 = 324x
12
A) 17
4
B) 3
36)
4
C) 17
D) - 2
Change the exponential expression to an equivalent expression involving a logarithm.
37) 6 3 = 216
A) log
35}
6
3 = 216
38) 4 5/2 = 32
log 32
2
=4
A)
log 4
5
B) log
3
216 = 6
C) log
216
6=3
D) log
6
216 = 3
38)
B) log
5
4=
5
2
C) log
4
32 =
5
2
D) log
32
4=
Change the logarithmic expression to an equivalent expression involving an exponent.
1
= -3
39) log
2 8
A) 2 -3 =
1
8
25 = x
5
A) x5 = 25
37)
1
C) 3 2 =
8
B) 2 8 = 3
5
2
39)
D)
1 3
=2
8
40) log
40)
B) 255 = x
C) 5 x = 25
6
D) 25x = 5
Find the exact value of the logarithmic expression.
41) log7 343
A) 21
42) log6
41)
B) 7
C) 343
D) 3
1
36
42)
A) -2
Find the domain of the function.
43) f(x) = ln(-7 - x)
A) (-7, )
44) f(x) = log10
B) 6
C) -6
B) (7, )
C) (
D) 2
, 7)
D) (
43)
, -7)
x+2
x-7
A) (- , -2)
(7, )
45) f(x) = ln x
A) (- , 0)
44)
B) (- , -2)
C) (7, )
D) (-2, 7)
B) (1, )
C) (- , 1)
D) (0, )
45)
SHOW YOUR WORK.
Graph the function.
46) f(x) = 1 - ln x
46)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
47) f(x) = log (x - 2)
5
47)
7
A)
B)
C)
D)
SHOW YOUR WORK.
48) f(x) = -2 + log
3
48)
x
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the equation.
49) log9 x2 = 4
A) {512}
50) log72 (x2 - x) = 1
A) {8, 9}
49)
B) {6, -6}
C) {6561}
D) {81, -81}
B) {1, 72}
C) {-8, 9}
D) {-8, -9}
50)
8
51) 7 ln 2x = 35
A)
5
ln 2
51)
e5
B)
2
C) {e5 }
D) e 5/2
SHOW YOUR WORK.
Write as the sum and/or difference of logarithms. Express powers as factors.
4
52) log
53) log
11
15 y2 x
xy
15
7
7
54) log
2
3
m n
k2
(x + 8)(x - 9) 4/5
,
(x - 5)4
55) ln
53)
x>9
55)
Express as a single logarithm.
56) 6 log m - log n
b
b
57)
56)
1
(log6 (x - 5) - log6 x)
2
58) 4 log4 2 +
57)
1
1
log4 (x - 6) - log4 x
6
2
58)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the Change-of-Base Formula and a calculator to evaluate the logarithm. Round your answer to three decimal places.
59) log 0.088
59)
6
A) -0.737
B) -1.056
C) 68.182
D) -1.356
60) log
0.057
2
A) -0.242
60)
B) -4.133
Solve the equation.
61) log (2 + x) - log (x - 4) = log 3
A) {-7}
62) log
6
B)
C) -1.244
61)
5
2
C) {7}
(4x + 3) = log (4x + 7)
6
A) {0}
D) 35.088
D)
62)
B) {2}
C)
9
5
2
D)
63) log2 (3x - 2) - log2 (x - 5) = 4
A) {18}
63)
B)
3
13
C) {6}
(x + 4) + log (x - 2) = 2
4
4
A) {-6}
B) {4, -6}
D)
38
5
64) log
65) 3 2x + 3 x - 6 = 0
A) {ln 6}
B)
64)
C) {4}
ln 3
ln 2
C)
ln 2
ln 6
D) {5}
D)
65)
ln 2
ln 3
SHOW YOUR WORK.
Solve the equation. Express irrational answers in exact form and as a decimal rounded to 3 decimal places.
2 x
= 71 - x
66)
66)
3
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
67) The size P of a small herbivore population at time t (in years) obeys the function P(t) = 900e0.22t if
they have enough food and the predator population stays constant. After how many years will the
population reach 1800? Round to the nearest hundredth.
A) 9.99 yr
B) 3.15 yr
C) 30.92 yr
D) 7.7 yr
67)
68) Conservationists tagged 150 black-nosed rabbits in a national forest in 2009. In 2010, they tagged
300 black-nosed rabbits in the same range. If the rabbit population follows the exponential law,
how many rabbits will be in the range 4 years from 2009?
A) 4800 rabbits
B) 357 rabbits
C) 2400 rabbits
D) 178 rabbits
68)
69) A fossilized leaf contains 9% of its normal amount of carbon 14. How old is the fossil (to the nearest
year)? Use 5600 years as the half-life of carbon 14.
A) 17,720
B) 761
C) 36,378
D) 19,419
69)
SHOW YOUR WORK.
70) The half-life of radium is 1690 years. If 150 grams is present now, how long (to the nearest
year) till only 100 grams are present?
70)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
71) The logistic growth model P(t) =
1
1 + 5.25e-0.821t
represents the proportion of the total market of a
new product as it penetrates the market t years after introduction. When will the product have 70%
of the market?
A) 4.05 years
B) 3.05 years
C) 1.59 years
D) 2.59 years
10
71)
Answer Key
1)
2)
3)
4)
5)
6)
7)
8)
D
D
C
B
A
D
C
A
30x
9)
5 + 6x
10) x
11) f(x) =
3
x ; g(x) = x + 1
1
12) f(x) = , g(x) = x2 - 5
x
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
f(x) = x ;
g(x) = 4x + 8
{x x -15}
{x x -3}
{x x 0, x 3}
{x 7 < x 8}
{x x 1, x 101}
{(-11, 8), (-10, 6), (-9, 4), (-8, 2)}; D = {-11, -10, -9, -8 }; R = {8, 6, 4, 2}
B
B
11
Answer Key
23)
5x + 2
; domain of f: {x x
24) f-1 (x) =
3-x
25)
26)
27)
28)
29)
-5}; range of f: {y y
D
B
C
B
domain of f: (- , ); range of f: (- , 4);
horizontal asymptote: y = 4
12
3}
Answer Key
30)
31)
32)
33)
34)
35)
36)
37)
38)
39)
40)
41)
42)
43)
44)
45)
domain of f: (- , ); range of f:(2, )
horizontal asymptote: y = 2
B
A
A
B
A
D
C
A
C
D
A
D
A
D
13
Answer Key
46)
47) D
48)
49) D
50) C
51) B
1
log 11 - 2 log y - log x
52)
15
15
15
4
53)
1
1
1
log x + log y - log 15
7
7
7
2
2
2
54)
1
1
log m + log n - 2 log k
2
2
2
7
3
55)
4
4
16
ln (x + 8) + ln (x - 9) ln (x - 5)
5
5
5
56) log
m6
b n
57) log6
58) log4
x-5
x
16
6
x-6
x
59) D
60) B
61) C
14
Answer Key
62)
63)
64)
65)
66)
67)
68)
69)
70)
71)
D
C
C
D
ln 7
2
ln
+ ln 7
3
1.263
B
C
D
989 years
B
15