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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