Math 1050 Exam 3 Review
6.1 Evaluate the expression using the values given in the table.1) (f∘g)(3)
x 1 7 11 12
f(x) -1 11 2 14
x -5 -1 1 3
g(x) 1 -6 7 11
Evaluate the expression using the graphs of y = f(x) and y = g(x).
2) Evaluate (fg)(1).
For the given functions f and g, find the requested composite function value.
3) f(x) = 2x + 4, g(x) = 4x2 + 3; Find (g ∘ f)(3).
1
For the given functions f and g, find the requested composite function.
4) f(x) = 5x + 13, g(x) = 5x - 1; Find (f ∘ g)(x).
5) f(x) = x - 6
5, g(x) = 5x + 6; Find (g ∘ f)(x).
Decide whether the composite functions, f ∘ g and g ∘ f, are equal to x.
6) f(x) = 6x, g(x) = x
6
7) f(x) = x + 1 , g(x) = x2
Find functions f and g so that f ∘ g = H.
8) H(x) = ∣9x + 10∣
2
9) H(x) = 1
x2 - 7
Solve the problem.
10) The surface area of a balloon is given by S(r) = 4πr2, where r is the radius of the balloon. If the radius is
increasing with time t, as the balloon is being blown up, according to the formula r(t) = 4
5t3, t ≥ 0, find the
surface area S as a function of the time t.
Find the domain of the composite function f ∘ g.
11) f(x) = 8x + 72; g(x) = x + 7
12) f(x) = 3
x + 3; g(x) = x + 8
13) f(x) = -3
x - 2; g(x) =
-16
x
3
14) f(x) = x; g(x) = 3x + 15
15) f(x) = 4
x - 6; g(x) = x - 4
6.2 Indicate whether the function is one-to-one.16) {(-13, -20), (-10, -20), (13, -8)}
Use the horizontal line test to determine whether the function is one-to-one.
17)
x
y
x
y
4
Find the inverse of the function and state its domain and range .
18) {(-3, 4), (-1, 5), (0, 2), (2, 6), (5, 7)}
Find the inverse. Determine whether the inverse represents a function.
19) {(6, -4), (2, -3), (0, -2), (-2, -1)}
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.
20) f(x) = x + 4
x-10 -5 5 10
y
10
5
-5
-10
x-10 -5 5 10
y
10
5
-5
-10
5
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.
21)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
(-4, -2)
(-2, 1)(0, 2)
(1, 4)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
(-4, -2)
(-2, 1)(0, 2)
(1, 4)
Decide whether or not the functions are inverses of each other.
22) f(x) = (x - 6)2, x ≥ 6; g(x) = x + 6
23) f(x) = (x - 3)2, x ≥ 3; g(x) = x + 3
24) f(x) = 5 + x
x, g(x) =
5
x - 1
6
The function f is one-to-one. Find its inverse.
25) f(x) = 6x + 4
26) f(x) = x2 + 5, x ≥ 0
27) f(x) = x + 8
Find the inverse function of f. State the domain and range of f.
28) f(x) = 3x - 2
x + 5
6.3 Approximate the value using a calculator. Express answer rounded to three decimal places.29) 42.6
7
30) 4.8π
31) 3 6
32) e-1.5
Determine whether the given function is exponential or not. If it is exponential, identify the value of the base a.
33)
x H(x)
-1 6
0 10
1 14
2 18
3 22
8
34)
x H(x)
-111
4
0 1
14
11
216
121
364
1331
Use transformations to graph the function. Determine the domain, range, and horizontal asymptote of the function.
35) f(x) = -2x+3 + 4
x-10 -5 5 10
y
10
5
-5
-10
x-10 -5 5 10
y
10
5
-5
-10
9
36) f(x) = 5-x + 2
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
Graph the function.
37) f(x) = 2(x + 2) - 2.
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
10
38) f(x) = ex
x-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
39) f(x) = 6 - e-x
x-10 -5 5 10
y
10
5
-5
-10
x-10 -5 5 10
y
10
5
-5
-10
Solve the equation.
40) 41 + 2x = 64
11
41) 5-x = 1
625
42) 2x2 - 3= 64
43) (ex)x · e45 = e14x
Solve the problem.
44) The number of books in a small library increases according to the function B = 8500e0.03t, where t is measured
in years. How many books will the library have after 2 years?
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6.4 Change the exponential expression to an equivalent expression involving a logarithm.
45) 5-2 = 1
25
46) ex = 7
Change the logarithmic expression to an equivalent expression involving an exponent.
47) log1/3
81 = -4
48) log4
16 = 2
49) ln x = 9
Find the exact value of the logarithmic expression.
50) log3 243
13
51) log4 1
64
52) log2
1
53) log3
3
54) ln e3
Use a calculator to evaluate the expression. Round your answer to three decimal places
55) log 5
6
56)log 9 + log 4
ln 3 - ln 6
14
Find the domain of the function.
57) f(x) = log(x - 4)
58) f(x) = log9(100 - x2)
Graph the function and its inverse on the same Cartesian plane.
59) f(x) = log5 x
x-5 5
y
5
-5
x-5 5
y
5
-5
15
Graph the function.
60) f(x) = 2 - ln(x + 4)
x-10 -5 5 10
y
10
5
-5
-10
x-10 -5 5 10
y
10
5
-5
-10
61) f(x) = log4
(x - 2)
x-10 -5 5 10
y
10
5
-5
-10
x-10 -5 5 10
y
10
5
-5
-10
Solve the equation.
62) log5
x = 2
16
63) log5 (x2 - 4x) = 1
64) 6 ln 9x = 12
65) e4x = 2
6.5 Use the properties of logarithms to find the exact value of the expression. Do not use a calculator.66) log4 416
67) ln e 5
68) log5 20 · log20 125
17
69) eln 5
Suppose that ln 2 = a and ln 5 = b. Use properties of logarithms to write each logarithm in terms of a and b.
70) ln 20
Write as the sum and/or difference of logarithms. Express powers as factors.
71) log18
11 r
s
72) log7
611
s2r
73) ln (5x)
91 + 3x
(x - 7)7 , x > 7
18
Express as a single logarithm.
74) 2 logb
9 + 10 logb
6
75) 5 logc
q - 2
3log
cr +
1
4log
cf - 3 log
cp
Use the Change-of-Base Formula and a calculator to evaluate the logarithm. Round your answer to two decimal places.
76) log7.7 3.2
77) log(2/3)19
6.6 Solve the equation.78) log
3x = 5
79) log (x + 2) = log (5x - 3)
19
80)1
3 log
2(x + 6) = log
8(3x)
81) log15
(x + 30) = 3 - log15
x
82) 2(7 - 3x) = 1
4
Solve the equation. Express irrational answers in exact form and as a decimal rounded to 3 decimal places.
83) 4 x = 61 - x
20
6.7 Find the amount that results from the investment.84) $480 invested at 10% compounded quarterly after a period of 8 years
Find the effective rate of interest.
85) 11.8% compounded continuously
86) 50.06% compounded daily
Find the present value. Round to the nearest cent.
87) To get $25,000 after 10 years at 11% compounded semiannually
Solve the problem.
88) What principal invested at 6%, compounded continuously for 3 years, will yield $1500? Round the answer to
two decimal places.
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Solve the problem. Round your answer to three decimals.
89) What annual rate of interest is required to triple an investment in 4 years?
Solve the problem.
90) How long does it take $1125 to triple if it is invested at 7% interest, compounded quarterly? Round your answer
to the nearest tenth.
91) 6.8 Conservationists tagged 50 black-nosed rabbits in a national forest in 1990. In 1992, they tagged 100black-nosed rabbits in the same range. If the rabbit population follows the exponential law, how many rabbits
will be in the range 6 years from 1990?
92) Assume that the half-life of Carbon-14 is 5700 years. Find the age (to the nearest year) of a wooden axe in
which the amount of Carbon-14 is 30% of what it originally had.
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93) A thermometer reading 34°F is brought into a room with a constant temperature of 80°F. If the thermometer
reads 45°F after 5 minutes, what will it read after being in the room for 7 minutes? Assume the cooling follows
Newton's Law of Cooling:
U = T + (Uo - T)ekt.
(Round your answer to two decimal places.)
94) The logistic growth model P(t) = 1490
1 + 28.8e-0.34t represents the population of a bacterium in a culture tube after
t hours. What was the initial amount of bacteria in the population?
95) The logistic growth model P(t) = 960
1 + 31e-0.337t represents the population of a bacterium in a culture tube after t
hours. When will the amount of bacteria be 800?
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