Deck 12: Exponential Functions and Logarithmic Functions

Question 1

Solve the problem.

-An accountant tabulated a firm's profits for four recent years in the following table:
$\begin{array} { l | l } \text { Year } & \text { Profits } \\\hline 1996 & \$ 250,000 \\1997 & \$ 300,000 \\1998 & \$ 400,000 \\1999 & \$ 600,000\end{array}$

The accountant then fit both a linear graph and an exponential curve (seen below) to the data, in order to estimat profits. Use the exponential graph to estimate the profits in the year $2002 .$

A) About $\$ 1,000,000$
B) About $\$ 1,700,000$
C) About $\$ 1,300,000$
D) About $\$ 750,000$

Answer 1

About $\$ 1,300,000$

Question 2

Solve the problem.
Suppose that $\$ 40,000$ is invested at $6 \%$ interest, compounded annually. Find a function A for the amount in the account after $t$ years.

A) $\mathrm { A } ( \mathrm { t } ) = \$ 40,000 ^ { 0.06 \mathrm { t } }$
B) $\mathrm { A } ( \mathrm { t } ) = \$ 40,000 ( 1.06 ) ^ { \mathrm { t } }$
C) $\mathrm { A } ( \mathrm { t } ) = \$ 40,000 ^ { 1.06 t }$
D) $\mathrm { A } ( \mathrm { t } ) = \$ 40,000 ( 0.06 ) ^ { t }$

Answer 2

B

Question 3

Solve the problem.

-An accountant tabulated a firm's profits for four recent years in the following table:
$\begin{array} { l | l } \text { Year } & \text { Profits } \\\hline 1996 & \$ 250,000 \\1997 & \$ 300,000 \\1998 & \$ 400,000 \\1999 & \$ 600,000\end{array}$

The accountant then fit both a linear graph and an exponential curve (seen below) to the data, in order to estimat profits. Use the exponential graph to estimate the profits in the year $2001 .$

A) About $\$ 750,000$
B) About $\$ 300,000$
C) About $\$ 1,300,000$
D) About $\$ 1,000,000$

Answer 3

About $\$ 1,000,000$

Question 4

Graph.
$$x = 2 y$$

A)

B)

C)

D)

Question 5

Solve the problem.
A computer is purchased for $\$ 4500$ . Its value each year is about $77 \%$ of the value the preceding year. Its value, in dollars, after $t$ years is given by the exponential function $V ( t ) = 4500 ( 0.77 ) ^ { t }$ . Find the value of the computer after 6 years.

A) $\$ 556.08$
B) $\$ 20,790.00$
C) $\$ 937.90$
D) $\$ 722.18$

Question 6

Solve the problem.
The amount of particulate matter left in solution during a filtering process decreases by the equation $\mathrm { P } ( \mathrm { n } ) = 700 ( 0.5 ) ^ { 0.6 \mathrm { n } }$ , where $\mathrm { n }$ is the number of filtering steps. Find the amounts left for $\mathrm { n } = 0$ and $\mathrm { n } = 5$ . (Round to the nearest whole number.)

A) $700 ; 22$
B) $700 ; 88$
C) $1400 ; 88$
D) $700 ; 5600$

Question 7

Solve the problem.

-An accountant tabulated a firm's profits for four recent years in the following table:
$\begin{array} { l | l } \text { Year } & \text { Profits } \\\hline 1996 & \$ 250,000 \\1997 & \$ 300,000 \\1998 & \$ 400,000 \\1999 & \$ 600,000\end{array}$

The accountant then fit both a linear graph and an exponential curve (seen below) to the data, in order to estimat profits. Use the linear graph to estimate the profits in the year $2002 .$

A) About $\$ 800,000$
B) About $\$ 500,000$
C) About $\$ 1,000,000$
D) About $\$ 900,000$

Question 8

Graph.
$f(x)=4^{x}$

A)

B)

C)

D)

Question 9

Graph.
$f(x)=\left(\frac{1}{3}\right)^{x}+2$

A)

B)

C)

D)

Question 10

Solve the problem.
The half-life of a certain radioactive substance is 8 years. Suppose that at time $t = 0$ , there are $29 \mathrm {~g}$ of the substance. Then after t years, the number of grams of the substance remaining will be:
$$N ( t ) = 29 \left( \frac { 1 } { 2 } \right) ^ { t / 16 }$$
How many grams of the substance will remain after 56 years? Round to the nearest hundredth when necessary.

A) $0.64 \mathrm {~g}$
B) $1.28 \mathrm {~g}$
C) $0.32 \mathrm {~g}$
D) $2.56 \mathrm {~g}$

Question 11

Solve the problem.
The half-life of Cesium $134 \mathrm {~m}$ is $3.0$ hours. If the formula $\mathrm { P } ( \mathrm { t } ) = \left( \frac { 1 } { 2 } \right) ^ { \mathrm { t } / 3.0 }$ gives the percent (as a decimal) remaining after time $\mathrm { t }$ (in hours), sketch P versus $\mathrm { t }$ .

A)

B)

C)

D)

Question 12

Graph.
$f(x)=2^{-x}$

A)

B)

C)

D)

Question 13

Solve the problem.
The number of bacteria growing in an incubation culture increases with time according to $\mathrm { B } ( \mathrm { x } ) = 9800 ( 2 ) ^ { \mathrm { x } }$ , where $x$ is time in days. Find the number of bacteria when $x = 0$ and $x = 3$ .

A) 19, $600 ; 78,400$
B) $9800 ; 58,800$
C) $9800 ; 39,200$
D) $9800 ; 78,400$

Question 14

Graph.
$x=\left(\frac{1}{4}\right)^{y}$

A)

B)

C)

D)

Question 15

Solve the problem.
The number of dislocated electric impulses per cubic inch in a transformer increases when lightning strikes by $\mathrm { D } ( \mathrm { x } ) = 1000 ( 2 ) ^ { \mathrm { x } }$ , where $\mathrm { x }$ is the time in milliseconds of the lightning strike. Find the number of dislocated impulses at $x = 0$ and $x = 5$ .

A) $2000 ; 32,000$
B) $1000 ; 10,000$
C) $1000 ; 32,000$
D) $1000 ; 4000$

Question 16

Graph.
$$f ( x ) = 3 ^ { 2 x - 1 }$$

A)

B)

C)

D)

Question 17

Graph.
$f(x)=3 x-1$

A)

B)

C)

D)

Question 18

Solve the problem.

-An accountant tabulated a firm's profits for four recent years in the following table:
$\begin{array} { l | l } \text { Year } & \text { Profits } \\\hline 1996 & \$ 250,000 \\1997 & \$ 300,000 \\1998 & \$ 400,000 \\1999 & \$ 600,000\end{array}$

The accountant then fit both a linear graph and an exponential curve (seen below) to the data, in order to estimat profits. Use the linear graph to estimate the profits in the year $2001 .$

A) About $\$ 500,000$
B) About $\$ 700,000$
C) About $\$ 800,000$
D) About $\$ 900,000$

Question 19

Graph.
$f(x)=\left(\frac{1}{2}\right)^{x}$

A)

B)

C)

D)

Question 20

Graph.
$f(x)=4^{x}-3$

A)

B)

C)

D)

Question 21

Find the requested composition of functions.
Given $f ( x ) = - 5 x + 2$ and $g ( x ) = 2 x + 4$ , find $g f ( x )$ .

A) $x + 8$
B) $- 10 x + 8$
C) $- 10 x + 22$
D) $x - 8$

Question 22

Find the inverse of the relation.
$$\{ ( 6,0 ) , ( - 4,1 ) , ( - 6,2 ) , ( - 8,3 ) \}$$

A) $\{ ( 1,0 ) , ( 3 , - 6 ) , ( 6 , - 6 ) , ( 1,2 ) \}$
B) $\{ ( 0,6 ) , ( 1 , - 4 ) , ( 2 , - 6 ) , ( 3 , - 8 ) \}$
C) $\{ ( 1,0 ) , ( 0 , - 6 ) , ( 6 , - 4 ) , ( 1,2 ) \}$

Question 23

Find f(x)and g(x)such that h(x)= (f ° g)(x).
$$h ( x ) = \frac { x ^ { 5 } - 4 } { x ^ { 5 } + 2 }$$

A) $f ( x ) = \frac { x - 4 } { x + 2 } , g ( x ) = x ^ { 5 }$
B) $f ( x ) = x ^ { 5 } + 2 , g ( x ) = x ^ { 5 } - 4$
C) $f ( x ) = x ^ { 5 } , g ( x ) = \frac { x - 4 } { x + 2 }$
D) $f ( x ) = \frac { 1 } { x ^ { 5 } + 2 } , g ( x ) = x ^ { 5 } - 4$

Question 24

Find f(x)and g(x)such that h(x)= (f ° g)(x).
$$h ( x ) = \frac { 6 } { \sqrt { 10 x + 7 } }$$

A) $f ( x ) = 6 , g ( x ) = \sqrt { 10 + 7 }$
B) $f ( x ) = \sqrt { 10 x + 7 } , g ( x ) = 6$
C) $f ( x ) = \frac { 6 } { \sqrt { x } } , g ( x ) = 10 x + 7$
D) $f ( x ) = \frac { 6 } { x } , g ( x ) = 10 x + 7$

Question 25

Find the inverse of the relation.
$$\{ ( - 16,3 ) , ( 5 , - 16 ) , ( - 14 , - 12 ) \}$$

A) $\{ ( - 16 , - 16 ) , ( - 16,5 ) , ( - 12 , - 14 ) \}$
B) $\{ ( 3 , - 16 ) , ( - 16,5 ) , ( - 12 , - 14 ) \}$
C) $\{ ( 3 , - 16 ) , ( - 14,5 ) , ( - 12 , - 16 ) \}$

Question 26

Find f(x)and g(x)such that h(x)= (f ° g)(x).
$$h ( x ) = \frac { 1 } { x ^ { 2 } - 5 }$$

A) $f ( x ) = \frac { 1 } { 5 } , g ( x ) = x ^ { 2 } - 5$
B) $f ( x ) = \frac { 1 } { x } , g ( x ) = x ^ { 2 } - 5$
C) $f ( x ) = \frac { 1 } { x ^ { 2 } } , g ( x ) = x - 5$
D) $f ( x ) = \frac { 1 } { x ^ { 2 } } , g ( x ) = - \frac { 1 } { 5 }$

Question 27

Find f(x)and g(x)such that h(x)= (f ° g)(x).
$$h ( x ) = \sqrt { 58 x ^ { 2 } + 92 }$$

A) $f ( x ) = \sqrt { 58 x ^ { 2 } } , g ( x ) = \sqrt { 92 }$
B) $f ( x ) = 58 x ^ { 2 } + 92 , g ( x ) = \sqrt { x }$
C) $f ( x ) = \sqrt { x } , g ( x ) = 58 x ^ { 2 } + 92$
D) $f ( x ) = \sqrt { 58 x + 92 } , g ( x ) = x ^ { 2 }$

Question 28

Find the requested composition of functions.
Given $f ( x ) = \frac { 6 } { x ^ { 2 } }$ and $g ( x ) = x - 3$ , find $g f ( x )$

A) $\frac { 6 } { x ^ { 2 } - 3 }$
B) $\frac { 6 } { x ^ { 2 } } - 3$
C) $\frac { 3 } { x ^ { 2 } }$
D) $\frac { 6 } { ( x - 3 ) ^ { 2 } }$

Question 29

Find f(x)and g(x)such that h(x)= (f ° g)(x).
$$h ( x ) = \frac { 8 } { x ^ { 2 } } + 7$$

A) $f ( x ) = \frac { 8 } { x ^ { 2 } } , g ( x ) = 7$
B) $f ( x ) = x , g ( x ) = \frac { 8 } { x } + 7$
C) $f ( x ) = x + 7 , g ( x ) = \frac { 8 } { x ^ { 2 } }$
D) $f ( x ) = \frac { 1 } { x } , g ( x ) = \frac { 8 } { x } + 7$

Question 30

Find the requested composition of functions.
Given $f ( x ) = \frac { x - 10 } { 7 }$ and $g ( x ) = 7 x + 10$ , find $g f ( x )$ .

A) $x - \frac { 10 } { 7 }$
B) $x + 20$
C) $7 x + 60$
D) $x$

Question 31

Find the inverse of the relation.
$$\{ ( - 5 , - 4 ) , ( 5,4 ) , ( - 3,6 ) , ( 3 , - 6 ) \}$$

A) $\{ ( - 4 , - 5 ) , ( 4,5 ) , ( 6,5 ) , ( - 6,3 ) \}$
B) $\{ ( - 4 , - 5 ) , ( 4,5 ) , ( 6 , - 3 ) , ( - 6,3 ) \}$
C) $\{ ( - 4 , - 5 ) , ( - 5,5 ) , ( 6 , - 3 ) , ( - 6,3 ) \}$

Question 32

Find the requested composition of functions.
Given $f ( x ) = x ^ { 2 } + 8$ and $g ( x ) = x ^ { 2 } - 8$ , find $f g ( x )$ .

A) $x ^ { 4 } - 16 x ^ { 2 } + 72$
B) $x ^ { 4 } + 16 x ^ { 2 } + 72$
C) $x ^ { 4 } + 16 x ^ { 2 } + 56$
D) $x ^ { 4 } - 16 x ^ { 2 } + 56$

Question 33

Find the requested composition of functions.
Given $f ( x ) = 2 x ^ { 2 } - 3$ and $g ( x ) = \frac { 3 } { x }$ , find $f g ( x )$ .

A) $\frac { 18 } { x ^ { 2 } } - 3$
B) $6 x - \frac { 9 } { x }$
C) $\frac { 3 } { 2 x ^ { 2 } - 3 }$
D) $\frac { 18 } { x } - 3$

Question 34

Find f(x)and g(x)such that h(x)= (f ° g)(x).
$$h ( x ) = | 7 x + 5 |$$

A) $f ( x ) = x , g ( x ) = 7 x + 5$
B) $f ( x ) = | - x | , g ( x ) = 7 x - 5$
C) $f ( x ) = - | x | , g ( x ) = 7 x + 5$
D) $f ( x ) = | x | , g ( x ) = 7 x + 5$

Question 35

Find the requested composition of functions.
Given $f ( x ) = \frac { 3 } { x }$ and $g ( x ) = 2 x ^ { 2 }$ , find $g f ( x )$ .

A) $\frac { 2 x ^ { 2 } } { 3 }$
B) $\frac { 2 x ^ { 2 } } { 9 }$
C) $\frac { 3 } { 2 x ^ { 2 } }$
D) $\frac { 18 } { \mathrm { x } ^ { 2 } }$

Question 36

Find the inverse of the relation.
$$\{ ( - 8 , - 7 ) , ( 7,8 ) , ( - 4 , - 3 ) , ( 4,3 ) \}$$

A) $\{ ( 3 , - 4 ) , ( - 4,7 ) , ( - 7 , - 8 ) , ( - 3,4 ) \}$
B) $\{ ( - 7 , - 8 ) , ( 8,7 ) , ( - 3 , - 4 ) , ( 3,4 ) \}$
C) $\{ ( 3 , - 4 ) , ( 8,7 ) , ( - 7,7 ) , ( - 3,4 ) \}$

Question 37

Find f(x)and g(x)such that h(x)= (f ° g)(x).
$$h ( x ) = ( 9 x + 9 ) ^ { 7 }$$

A) $f ( x ) = ( 9 x ) ^ { 7 } , g ( x ) = 9$
B) $f ( x ) = 9 x + 9 , g ( x ) = x ^ { 7 }$
C) $f ( x ) = 9 x ^ { 7 } , g ( x ) = x + 9$
D) $f ( x ) = x ^ { 7 } , g ( x ) = 9 x + 9$

Question 38

Find the requested composition of functions.
Given $f ( x ) = 7 x + 8$ and $g ( x ) = 5 x - 1$ , find $f g ( x )$ .

A) $35 x + 39$
B) $35 x + 15$
C) $35 x + 1$
D) $35 x + 7$

Question 39

Find f(x)and g(x)such that h(x)= (f ° g)(x).
$$h ( x ) = 6 ( 8 x + 7 ) ^ { 2 } - 9$$

A) $f ( x ) = 8 x + 7 , g ( x ) = 6 x ^ { 2 } - 9$
B) $f ( x ) = 6 x ^ { 2 } - 9 , g ( x ) = ( 8 x + 7 ) ^ { 2 }$
C) $f ( x ) = 6 x ^ { 2 } - 9 , g ( x ) = 8 x + 7$
D) $f ( x ) = ( 6 x - 9 ) ^ { 2 } , g ( x ) = 8 x + 7$

Question 40

Find f(x)and g(x)such that h(x)= (f ° g)(x).
$$h ( x ) = ( \sqrt { x } - 5 ) ^ { 5 }$$

A) $f ( x ) = x ^ { 5 } , g ( x ) = \sqrt { x } - 5$
B) $f ( x ) = \sqrt { x ^ { 5 } } , g ( x ) = x - 5$
C) $f ( x ) = \sqrt { x } - 5 , g ( x ) = x ^ { 5 }$
D) $f ( x ) = \sqrt { x } , g ( x ) = ( x - 5 ) ^ { 5 }$

Question 41

Graph the relation using solid circles and the inverse using open circles.
$$\{ ( - 9,3 ) , ( 9 , - 3 ) , ( - 7 , - 1 ) , ( 7,1 ) \}$$

A)

B)

C)

D)

Question 42

Graph the relation using solid circles and the inverse using open circles.
$$\{ ( - 1,16 ) , ( - 7 , - 2 ) , ( 5 , - 11 ) \}$$

A)

B)

C)

D)

Question 43

Graph the relation using solid circles and the inverse using open circles.
$$\{ ( - 3 , - 12 ) , ( - 5 , - 11 ) , ( - 7 , - 10 ) , ( - 9 , - 9 ) \}$$

A)

B)

C)

D)

Question 44

Graph the equation of the relation using a solid line, and then graph the inverse of the relation using a dashed line.
$$y = 7 x ^ { 2 } + 2 x$$

A)

B)

C)

D)

Question 45

Find an equation of the inverse of the relation.
$$y = 3 x ^ { 2 } + 5 x$$

A) $x = 5 y ^ { 2 } + 3 y$
B) $y = 5 x ^ { 2 } + 3 x$
C) $x = 3 y ^ { 2 } + 5 y$
D) $y = - 3 x ^ { 2 } - 5 x$

Question 46

Determine whether the function is one-to-one.

- $$f ( x ) = 4 x ^ { 2 } + x$$

Question 47

Determine whether the function is one-to-one.

- $$f ( x ) = 6 x - 6$$

Question 48

Determine whether the function is one-to-one.

- $$f ( x ) = x ^ { 2 } - 6$$

Question 49

Find an equation of the inverse of the relation.
$$y = 2 x ^ { 3 } + 6$$

A) $y = - 2 x ^ { 3 } - 6$
B) $x = 2 y ^ { 3 } + 6$
C) $x = 6 y ^ { 3 } + 2$
D) $y = 6 x ^ { 3 } + 2$

Question 50

Determine whether the function is one-to-one.

- $$f ( x ) = 6 x ^ { 3 } - 2$$

Question 51

Determine whether the function is one-to-one.

- $$f ( x ) = 7 x ^ { 2 } - 6$$

Question 52

Determine whether the function is one-to-one.

- $$f ( x ) = x ^ { 3 } + 8$$

Question 53

Graph the equation of the relation using a solid line, and then graph the inverse of the relation using a dashed line.
$$y = 2 x ^ { 3 } + 3$$

A)

B)

C)

D)

Question 54

Find an equation of the inverse of the relation.
$$y = 4 + 6 x$$

A) $y = 6 - 4 x$
B) $y = 6 + 4 x$
C) $x = 6 + 4 y$
D) $x = 4 + 6 y$

Question 55

Determine whether the function is one-to-one.

- $$f ( x ) = \left| 36 - x ^ { 2 } \right|$$

Question 56

Graph the equation of the relation using a solid line, and then graph the inverse of the relation using a dashed line.
$$y = 7 x - 8$$

A)

B)

C)

D)

Question 57

Determine whether the function is one-to-one.

- $$f ( x ) = 64 - x ^ { 2 }$$

Question 58

Graph the equation of the relation using a solid line, and then graph the inverse of the relation using a dashed line.
$$y = 2 + 3 x$$

A)

B)

C)

D)

Question 59

Graph the relation using solid circles and the inverse using open circles.
$$\{ ( - 6,7 ) , ( - 7,6 ) , ( - 9 , - 4 ) , ( 9,4 ) \}$$

A)

B)

C)

D)

Question 60

Find an equation of the inverse of the relation.
$$y = 2 x - 7$$

A) $y = - 2 x + 7$
B) $y = - 7 x + 2$
C) $x = 2 y - 7$
D) $x = - 7 y + 2$

Question 61

Determine whether the function is one-to-one.

- $$f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x }$$

Question 62

Determine whether the given function is one-to-one. If so, find a formula for the inverse.
$$f ( x ) = 6 x ^ { 3 } + 4$$

A) $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { \frac { x - 4 } { 6 } }$
B) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sqrt [ 3 ] { \frac { \mathrm { x } + 4 } { 6 } }$
C) Not a one-to-one function
D) $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { \frac { x } { 6 } } - 4$

Question 63

Determine whether the given function is one-to-one. If so, find a formula for the inverse.
$$f ( x ) = \frac { 4 } { x + 5 }$$

A) $f ^ { - 1 } ( x ) = \frac { - 5 x + 4 } { x }$
B) $f ^ { - 1 } ( x ) = \frac { 5 + 4 x } { x }$
C) Not a one-to-one function
D) $f ^ { - 1 } ( x ) = \frac { x } { 5 + 4 x }$

Question 64

Determine whether the given function is one-to-one. If so, find a formula for the inverse.
$$f ( x ) = 4 x + 3$$

A) $f ^ { - 1 } ( x ) = \frac { x + 3 } { 4 }$
B) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { \mathrm { x } - 3 } { 4 }$
C) $f ^ { - 1 } ( x ) = \frac { x } { 4 } - 3$
D) Not a one-to-one function

Question 65

Determine whether the given function is one-to-one. If so, find a formula for the inverse.
$$f ( x ) = x ^ { 3 } - 8$$

A) $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { x } + 8$
B) $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { x + 8 }$
C) Not a one-to-one function
D) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sqrt [ 3 ] { \mathrm { x } - 8 }$

Question 66

Determine whether the given function is one-to-one. If so, find a formula for the inverse.
$$f ( x ) = ( x + 5 ) ^ { 2 }$$

A) $f ^ { - 1 } ( x ) = \sqrt { x - 5 }$
B) $f ^ { - 1 } ( x ) = \sqrt { x } - 5$
C) Not a one-to-one function
D) $f ^ { - 1 } ( x ) = \frac { 1 } { \sqrt { x - 5 } }$

Question 67

Determine whether the given function is one-to-one. If so, find a formula for the inverse.
$$f ( x ) = \sqrt [ 3 ] { x + 3 }$$

A) $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { x - 3 }$
B) Not a one-to-one function
C) $f ^ { - 1 } ( x ) = x ^ { 3 } - 3$
D) $f ^ { - 1 } ( x ) = ( x + 3 ) ^ { 3 }$

Question 68

Determine whether the given function is one-to-one. If so, find a formula for the inverse.
$$f ( x ) = \frac { 4 x - 7 } { 2 x + 6 }$$

A) Not a one-to-one function
B) $f ^ { - 1 } ( x ) = \frac { 4 x - 7 } { 2 x + 6 }$
C) $f ^ { - 1 } ( x ) = \frac { - 6 x - 7 } { 2 x - 4 }$
D) $f ^ { - 1 } ( x ) = \frac { 2 x - 4 } { - 6 x - 7 }$