Deck 5: Inverse, Exponential, and Logarithmic Functions

Question 1

Determine the range of for the function without actually finding . Hint: First find the domain and range of

A)
B)
C)
D)
E)

Answer 1

Question 2

Express in terms of logarithms of x, y, or z.

A)
B)
C)
D)
E)

Answer 2

Question 3

Graph f on the given interval. Estimate the largest interval [ a , b ] with a < 0 < b on which f is one-to-one.

A)
B)
C)
D)
E)

Answer 3

Question 4

Solve the equation.

A)18
B)7
C)1
D)9
E)8

Question 5

When the volume control on a stereo system is increased, the voltage across a loudspeaker changes from V ₁ to V ₂ , and the decibel increase is given by Find the decibel increase if the voltage changes from 1.7 volts to 4 volts.

A)db = 17.1
B)db = 6.3
C)db = 11.2
D)db = 7.4
E)db = 9.7

Question 6

Simplify the expression.

A)1
B)
C)
D)
E)

Question 7

Solve the equation:

A)
B)
C)
D)
E)

Question 8

Find an exponential function of the form f (x) = ba ^x that has the given y-intercept and passes through the point P: y-intercept = 125; P (3,1);

A)
B)
C)
D)
E)

Question 9

In fishery science, a cohort is the collection of fish that results from one annual reproduction. It is usually assumed that the number of fish N( t ) still alive after t years is given by an exponential function. For Pacific halibut, , where N _o is the initial size of the cohort. Approximate the percentage of the original number still alive after 9 years.

A)5.5%
B)50.4%
C)33.1%
D)16.5%
E)18.0%

Question 10

Change to exponential form.

A)
B)
C)
D)
E)

Question 11

Solve the equation:

A)
B)
C)
D)
E)

Question 12

An urban density model is a formula that relates the population density D ( in thousands/ mi ² ) to the distance x (in miles) from the center of the city. The formula for the central density a and coefficient of decay b has been found to be appropriate for many large U.S. cities. For the city of Atlanta in 1970, a = 5.5 and b = 0.10. At approximately what distance was the population density 2,800 per square mile?

A)2.9 mi
B)6.8 mi
C)14.7 mi
D)9.1 mi
E)62.3 mi

Question 13

Sketch the graph of the function.

A)
B)
C)
D)

Question 14

Solve the equation.

A)
B)
C)
D)
E)

Question 15

If $900 is deposited in a savings account that pays interest at a rate of % per year compounded continuously, find the balance after 7 years.

A)$1,393.95
B)$1,369.77
C)$3,484.87
D)$581.08
E)$1,418.56

Question 16

If 7 grams of salt is added to a quantity of water, then the amount that is undissolved after t minutes is given by: Sketch a graph that shows the value at any time from t = 0 to t = 10.

A)
B)
C)
D)

Question 17

The Gompertz function, is sometimes used to describe the sales of a new product whose sales are initially large but then level off toward a maximum saturation level. Graph the line and the Gompertz function with:

A)
B)
C)
D)

Question 18

Ventilation is an effective way to improve indoor air quality. In nonsmoking restaurants, air circulation requirements (in ) are given by the function , where x is the number of people in the dining area. Find . Use to determine the maximum number of people that should be in a restaurant having a ventilation capability of 2,700 .

A)72
B)73
C)77
D)79
E)78

Question 19

If monthly payments p are deposited in a savings account paying an annual interest rate r, then the amount A in the account after n years is given by: Estimate n for p = 90, r = 0.08 and A = $259,153.

A)
B)
C)
D)
E)

Question 20

If the pollution of Lake Erie were stopped suddenly, it has been estimated that the level y of pollutants would decrease according to the formula y = y _o e ^-0.3937t , where t is the time in years and y _o is the pollutant level at which further pollution ceased. How many years would it take to clear 59% of the pollutants?

A)t = 0.98 years
B)t = 1.34 years
C)t = 2.02 years
D)t = 2.26 years
E)t = 0.58 years

Question 21

Solve the equation.

A)8
B)1
C)16
D)4
E)9

Question 22

The Gompertz function, is sometimes used to describe the sales of a new product whose sales are initially large but then level off toward a maximum saturation level. Graph the line and the Gompertz function with:

A)
B)
C)
D)

Question 23

Graph f on the given interval. Estimate the largest interval [ a , b ] with a < 0 < b on which f is one-to-one.

A)
B)
C)
D)
E)

Question 24

Simplify the expression.

A)1
B)
C)
D)
E)

Question 25

Sketch the graph of the function.

A)
B)
C)
D)

Question 26

If 3 grams of salt is added to a quantity of water, then the amount that is undissolved after t minutes is given by: Sketch a graph that shows the value at any time from t = 0 to t = 10.

A)
B)
C)
D)

Question 27

Use the change of base formula to approximate the x-intercept.

A)2.3219
B)0.0602
C)0.9027
D)0.4307
E)5.1603

Question 28

Determine the range of for the function without actually finding . Hint: First find the domain and range of

A)
B)
C)
D)
E)

Question 29

Solve the equation:

A)
B)
C)
D)
E)

Question 30

If monthly payments p are deposited in a savings account paying an annual interest rate r, then the amount A in the account after n years is given by: Estimate n for p = 98, r = 0.02 and A = $208,744.

A)
B)
C)
D)
E)

Question 31

If $1,200 is deposited in a savings account that pays interest at a rate of % per year compounded continuously, find the balance after 4 years.

A)$1,826.35
B)$1,808.18
C)$4,520.45
D)$1,790.19
E)$796.38

Question 32

An urban density model is a formula that relates the population density D ( in thousands/ mi ² ) to the distance x (in miles) from the center of the city. The formula for the central density a and coefficient of decay b has been found to be appropriate for many large U.S. cities. For the city of Atlanta in 1970, a = 5.5 and b = 0.10. At approximately what distance was the population density 1,800 per square mile?

A)13.5 mi
B)11.2 mi
C)57.9 mi
D)4.9 mi
E)23.5 mi

Question 33

In fishery science, a cohort is the collection of fish that results from one annual reproduction. It is usually assumed that the number of fish N( t ) still alive after t years is given by an exponential function. For Pacific halibut, , where N _o is the initial size of the cohort. Approximate the percentage of the original number still alive after 14 years.

A)6.1%
B)28.0%
C)2.0%
D)12.2%
E)81.8%

Question 34

Ventilation is an effective way to improve indoor air quality. In nonsmoking restaurants, air circulation requirements (in ) are given by the function , where x is the number of people in the dining area. Find . Use to determine the maximum number of people that should be in a restaurant having a ventilation capability of 2,550 .

A)81
B)75
C)82
D)80
E)76

Question 35

Find the exact solution, using common logarithms, and an approximation of the solution of the equation to two decimal places.

A)
B)
C)
D)
E)

Question 36

One method of estimating the thickness of the ozone layer is to use the formula , where I _o is the intensity of a particular wavelength of light from the sun before it reaches the atmosphere, I is the intensity of the same wavelength after passing through a layer of ozone x centimeters thick, and k is the absorption constant of ozone for that wavelength. Suppose a wavelength of cm with k = 0.43 , is measured as 1.17. Approximate the thickness of the ozone layer to the nearest 0.01 centimeter.

A)x = 2.72 cm
B)x = 3.09 cm
C)x = 0.24 cm
D)x = 0.37 cm
E)x = 0.07 cm

Question 37

Change to exponential form.

A)
B)
C)
D)
E)

Question 38

Find an exponential function of the form f (x) = ba ^x that has the given y-intercept and passes through the point P: y-intercept = 64; P (3,1);

A)
B)
C)
D)
E)

Question 39

Use common logarithms to solve for x in terms of y.

A)
B)
C)
D)
E)

Question 40

Solve the equation:

A)
B)
C)
D)
E)

Question 41

If $900 is deposited in a savings account that pays interest at a rate of % per year compounded continuously, find the balance after 7 years.

A)$1,418.56
B)$581.08
C)$1,369.77
D)$1,393.95
E)$3,484.87

Question 42

When the volume control on a stereo system is increased, the voltage across a loudspeaker changes from V ₁ to V ₂ , and the decibel increase is given by Find the decibel increase if the voltage changes from 2.2 volts to 3.5 volts.

A)db = 9.3
B)db = 2.9
C)db = 6.3
D)db = 7.8
E)db = 4.0

Question 43

Ventilation is an effective way to improve indoor air quality. In nonsmoking restaurants, air circulation requirements (in ) are given by the function , where x is the number of people in the dining area. Find . Use to determine the maximum number of people that should be in a restaurant having a ventilation capability of 2,350 .

A)55
B)56
C)62
D)68
E)61

Question 44

If 7 grams of salt is added to a quantity of water, then the amount that is undissolved after t minutes is given by: Sketch a graph that shows the value at any time from t = 0 to t = 10.

A)
B)
C)
D)

Question 45

Express in terms of logarithms of x, y, or z.

A)
B)
C)
D)
E)

Question 46

One method of estimating the thickness of the ozone layer is to use the formula , where I _o is the intensity of a particular wavelength of light from the sun before it reaches the atmosphere, I is the intensity of the same wavelength after passing through a layer of ozone x centimeters thick, and k is the absorption constant of ozone for that wavelength. Suppose a wavelength of cm with k = 0.45 , is measured as 1.14. Approximate the thickness of the ozone layer to the nearest 0.01 centimeter.

A)x = 0.06 cm
B)x = 2.82 cm
C)x = 0.16 cm
D)x = 2.53 cm
E)x = 0.29 cm

Question 47

Sketch the graph of the function.

A)
B)
C)
D)

Question 48

Graph f on the given interval. Estimate the largest interval [ a , b ] with a < 0 < b on which f is one-to-one.

A)
B)
C)
D)
E)

Question 49

Simplify the expression.

A)
B)1
C)
D)
E)

Question 50

The Gompertz function, is sometimes used to describe the sales of a new product whose sales are initially large but then level off toward a maximum saturation level. Graph the line and the Gompertz function with:

A)
B)
C)
D)

Question 51

Find the exact solution, using common logarithms, and an approximation of the solution of the equation to two decimal places.

A)
B)
C)
D)
E)

Question 52

Find an exponential function of the form f (x) = ba ^x that has the given y-intercept and passes through the point P: y-intercept = 125; P (3,1);

A)
B)
C)
D)
E)

Question 53

Determine the range of for the function without actually finding . Hint: First find the domain and range of

A)
B)
C)
D)
E)

Question 54

Use common logarithms to solve for x in terms of y.

A)
B)
C)
D)
E)

Question 55

Use the change of base formula to approximate the x-intercept.

A)2.8074
B)0.0430
C)1.6775
D)5.0672
E)0.3562

Question 56

If the pollution of Lake Erie were stopped suddenly, it has been estimated that the level y of pollutants would decrease according to the formula y = y _o e ^-0.3937t , where t is the time in years and y _o is the pollutant level at which further pollution ceased. How many years would it take to clear 59% of the pollutants?

A)t = 0.58 years
B)t = 1.34 years
C)t = 2.02 years
D)t = 0.98 years
E)t = 2.26 years

Question 57

Solve the equation:

A)
B)
C)
D)
E)

Question 58

If monthly payments p are deposited in a savings account paying an annual interest rate r, then the amount A in the account after n years is given by: Estimate n for p = 96, r = 0.04 and A = $177,664.

A)
B)
C)
D)
E)

Question 59

Solve the equation.

A)
B)
C)
D)
E)

Question 60

Solve the equation:

A)
B)
C)
D)
E)

Question 61

Express in terms of logarithms of x, y, or z.

A)
B)
C)
D)
E)

Question 62

Solve the equation.

A)1
B)7
C)2
D)5
E)4

Question 63

Solve the equation:

A)
B)
C)
D)
E)

Question 64

Use the change of base formula to approximate the x-intercept.

A)2.8074
B)0.3562
C)1.1363
D)6.1496
E)0.0430

Question 65

The graph of a one-to-one function f is shown. Use the reflection property to sketch the graph of

A)
B)
C)
D)

Question 66

When the volume control on a stereo system is increased, the voltage across a loudspeaker changes from V ₁ to V ₂ , and the decibel increase is given by Find the decibel increase if the voltage changes from 1.7 volts to 3 volts.

A)db = 4.9
B)db = 8.7
C)db = 7.2
D)db = 11.4
E)db = 3.8

Question 67

Use common logarithms to solve for x in terms of y.

A)
B)
C)
D)
E)

Question 68

Change to exponential form.

A)
B)
C)
D)
E)

Question 69

One method of estimating the thickness of the ozone layer is to use the formula , where I _o is the intensity of a particular wavelength of light from the sun before it reaches the atmosphere, I is the intensity of the same wavelength after passing through a layer of ozone x centimeters thick, and k is the absorption constant of ozone for that wavelength. Suppose a wavelength of cm with k = 0.36 , is measured as 1.12. Approximate the thickness of the ozone layer to the nearest 0.01 centimeter.

A)x = 0.31 cm
B)x = 3.43 cm
C)x = 0.04 cm
D)x = 3.11 cm
E)x = 0.18 cm

Question 70

Solve the equation.

A)
B)
C)
D)
E)

Question 71

Some lending institutions calculate the monthly payment M on a loan of L dollars at an interest rate r (expressed as a decimal) by using the formula: where k = [1 + ( r / 12 )] ^12t and t is the number of years that the loan is in effect. An automobile dealer offers customers no-down-payment 7-year loans at an interest rate of 7%. If a customer can afford to pay $108 per month, find the price of the most expensive car that can be purchased.

A)$ 7,145.44
B)$ 7,155.79
C)$ 7,161.43
D)$ 7,165.81
E)$ 7,164.60

Question 72

Find the exact solution, using common logarithms, and an approximation of the solution of the equation to two decimal places.

A)
B)
C)
D)
E)

Question 73

Find the inverse function of .

A)
B)
C)
D)
E)

Question 74

If the pollution of Lake Erie were stopped suddenly, it has been estimated that the level y of pollutants would decrease according to the formula y = y _o e ^-0.3937t , where t is the time in years and y _o is the pollutant level at which further pollution ceased. How many years would it take to clear 59% of the pollutants?

A)t = 0.58 years
B)t = 1.34 years
C)t = 2.02 years
D)t = 0.98 years
E)t = 2.26 years

Question 75

In fishery science, a cohort is the collection of fish that results from one annual reproduction. It is usually assumed that the number of fish N( t ) still alive after t years is given by an exponential function. For Pacific halibut, , where N _o is the initial size of the cohort. Approximate the percentage of the original number still alive after 7 years.

A)24.7%
B)26.0%
C)8.2%
D)49.3%
E)14.0%

Question 76

The graph of a one-to-one function f is shown. Find the domain D and range R of the inverse function .

A)
B)
C)
D)
E)

Question 77

Use the theorem on inverse functions to determine whether f and g are inverse functions of each other.

A)f and g are not inverse functions of each other.
B)f and g are inverse functions of each other.

Question 78

Find an exponential function of the form f (x) = ba ^x that has the given y-intercept and passes through the point P: y-intercept = 27; P (3,1);

A)
B)
C)
D)
E)

Question 79

An urban density model is a formula that relates the population density D ( in thousands/ mi ² ) to the distance x (in miles) from the center of the city. The formula for the central density a and coefficient of decay b has been found to be appropriate for many large U.S. cities. For the city of Atlanta in 1970, a = 5.5 and b = 0.10. At approximately what distance was the population density 800 per square mile?

A)39.8 mi
B)19.3 mi
C)21.6 mi
D)8.4 mi
E)49.8 mi

Question 80

The radioactive bismuth isotope ²¹⁰ Bi has a half-life of 5 days. If there is 100 milligrams of ²¹⁰ Bi present at t = 0, then the amount f(t) remaining after t days is given by: How much ²¹⁰ Bi remains after 5 days?

A)52 milligrams
B)55 milligrams
C)50 milligrams
D)49 milligrams
E)59 milligrams