Question 1

Use the definition of inverses to determine whether f and g are inverses.
-$f ( x ) = 7 x - 7 , \quad g ( x ) = \frac { 1 } { 7 } x + 1$

Accepted Answer

A) True 
 B)False

Question 2

A certain radioactive isotope has a half-life of approximately 1500 years. How many years to the nearest year would be required for a given amount of this isotope to decay to 70% of that amount?

Accepted Answer

A)  2605 yr 
B)  772 yr 
C)  702 yr 
D)  450 yr 
A)  2605 yr 
B)  772 yr 
C)  702 yr 
D)  450 yr

Question 3

Given $f ( x ) = \log _ { 2 } x$, evaluate the following.
(a) $\mathrm { f } \left( 2 ^ { 5 } \right)$
(b) $f \left( 2 \log _ { 2 } 2 \right)$
(c) $\mathrm { f } \left( 2 ^ { 2 } \log _ { 2 } 2 \right)$

Accepted Answer

The answer of Given \(f ( x ) = \log...

Question 4

Suppose the government wants to impose a tax on fossil fuels to reduce carbon emissions. The cost benefit is modeled by $\ln ( 1 - \mathrm { P } ) = - 0.0039 - 0.0048 \mathrm { x }$, where $\mathrm { x }$ represents the dollars of tax per ton of carbon emitted and $\mathrm { P }$ represents the percent reduction in emissions of carbon. ( $P$ is in decimal form.) Determine $P$ when $x = 52$. Round to three decimal places.

Accepted Answer

A)  $- 0.19$ 
B)  $0.224$ 
C)  $1.776$ 
D)  $- 0.218$ 
A)  $- 0.19$ 
B)  $0.224$ 
C)  $1.776$ 
D)  $- 0.218$

Question 5

Solve the following graphically. If necessary, round answers to the nearest thousandth.
-$$\log x ^ { 3 } = ( \log x ) ^ { 3 }$$

Accepted Answer

A)  {1, 3} 
B)  {1, 0.138} 
C)  {0, 1} 
D)  {1, 0.019, 53.957} 
A)  {1, 3} 
B)  {1, 0.138} 
C)  {0, 1} 
D)  {1, 0.019, 53.957}

Question 6

Write an equivalent expression in exponential form.
-$\log_{ ( x - 6 )} 18 = 1$

Accepted Answer

A)  $\{ 12 \}$ 
B)  $\{ - 24 \}$ 
C)  $\{ 24 \}$ 
D)  $\{ - 12 \}$ 
A)  $\{ 12 \}$ 
B)  $\{ - 24 \}$ 
C)  $\{ 24 \}$ 
D)  $\{ - 12 \}$

Question 7

Find the future value.
-$3169 invested for 4 years at 3% compounded annually

Accepted Answer

A)  $3571.44 
B)  $3569.86 
C)  $3566.74 
D)  $3572.50 
A)  $3571.44 
B)  $3569.86 
C)  $3566.74 
D)  $3572.50

Question 8

Round your answer to the nearest tenth, when appropriate. Use the formula $\mathrm { pH } = - \log \left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right]$ , as
needed.
-Find the pH if $\left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right] = 7.9 \times 10 ^ { - 3 }$ .

Accepted Answer

A)  3.1 
B)  3.9 
C)  2.9 
D)  2.1 
A)  3.1 
B)  3.9 
C)  2.9 
D)  2.1

Question 9

Round to the nearest thousandth.
-$$4 \mathrm { e } ^ { ( 3 \mathrm { x } + 2 ) } = 2$$

Accepted Answer

A)  $\{ - 0.898 \}$ 
B)  $\{ 0.292 \}$ 
C)  $\{ - 2.088 \}$ 
D)  $\{ - 1.626 \}$ 
A)  $\{ - 0.898 \}$ 
B)  $\{ 0.292 \}$ 
C)  $\{ - 2.088 \}$ 
D)  $\{ - 1.626 \}$

Question 10

In the formula N $\mathrm { N } = \mathrm { Ie }^ { kt } , \mathrm { N }$ N is the number of items in terms of an initial population I at a given time t and k is a growth constant equal to the percent of growth per unit time. There are currently 66 million cars in a certain
Country, increasing by 0.7% annually. How many years will it take for this country to have 87 million cars?
Round to the nearest year.

Accepted Answer

A)  39 yr 
B)  435 yr 
C)  5 yr 
D)  30 yr 
A)  39 yr 
B)  435 yr 
C)  5 yr 
D)  30 yr

Question 11

Use the alphabet coding method $$( A = 1 , B = 2 , C = 3 , \ldots Z = 26 )$$ to solve the problem.
-The function defined by $f ( x ) = x + 2$ was used to encode a message as: $\begin{array} { l l l l l l l l l l } 15 & 3 & 22 & 10 & 9 & 7 & 16 & 11 & 23 & 21 \end{array}$ Find the inverse function and determine the message.

Accepted Answer

The answer of Use the alphabet coding method \[( A...

Question 12

If the function is one-to-one, find its inverse. If not, write "not one-to-one."
-$$f ( x ) = - \sqrt { x ^ { 2 } - 25 } , x \geq 5$$

Accepted Answer

A)  $f ^ { - 1 } ( x ) = - \sqrt { 25 - x ^ { 2 } } , x \leq 0$ 
B)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sqrt { 25 - \mathrm { x } ^ { 2 } } , \mathrm { x } \leq 5$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sqrt { \mathrm { x } ^ { 2 } + 25 } , \mathrm { x } \geq 0$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sqrt { \mathrm { x } ^ { 2 } + 25 } , \mathrm { x } \leq 0$ 
A)  $f ^ { - 1 } ( x ) = - \sqrt { 25 - x ^ { 2 } } , x \leq 0$ 
B)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sqrt { 25 - \mathrm { x } ^ { 2 } } , \mathrm { x } \leq 5$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sqrt { \mathrm { x } ^ { 2 } + 25 } , \mathrm { x } \geq 0$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sqrt { \mathrm { x } ^ { 2 } + 25 } , \mathrm { x } \leq 0$

Question 13

Determine whether the statement is true or false.
-If a function f has an inverse and f(1) $f ( 1 ) = - 3 , \text { then } f ^ { - 1 } ( - 3 ) = 1$

Accepted Answer

A) True 
 B)False

Question 14

Solve the equation.
-$$\left( \frac { 6 } { 7 } \right) ^ { x } = \frac { 2401 } { 1296 }$$

Accepted Answer

A)  $\{ 4 \}$ 
B)  $\left\{ \frac { 1 } { 4 } \right\}$ 
C)  $\{ - 4 \}$ 
D)  $\left\{ - \frac { 1 } { 4 } \right\}$ 
A)  $\{ 4 \}$ 
B)  $\left\{ \frac { 1 } { 4 } \right\}$ 
C)  $\{ - 4 \}$ 
D)  $\left\{ - \frac { 1 } { 4 } \right\}$

Question 15

For the function as defined that is one-to-one, graph f and $\mathrm { f } ^ { - 1 }$ on the same axes.
-$f(x)=2 x$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 16

Determine whether the statement is true or false.
-If a function f has an inverse function, then f is one-to-one.

Accepted Answer

A) True 
 B)False

Question 17

Solve the equation.
-

Accepted Answer

A)  $- 3 ^ { x } + 1$ 
B)  $3 ^ { - x } + 1$ 
C)  $- 3 ^ { - x } + 1$ 
D)  $- 3 ^ { - x } - 1$ 
A)  $- 3 ^ { x } + 1$ 
B)  $3 ^ { - x } + 1$ 
C)  $- 3 ^ { - x } + 1$ 
D)  $- 3 ^ { - x } - 1$

Question 18

Find the value. Give an approximation to four decimal places.
-ln 0.984

Accepted Answer

A)  $0.0070$ 
B)  $0.0161$ 
C)  $- 0.0161$ 
D)  $- 0.0070$ 
A)  $0.0070$ 
B)  $0.0161$ 
C)  $- 0.0161$ 
D)  $- 0.0070$

Question 19

Quail are game birds that fare poorly when their habitat is encroached upon. Wildlife biologists have discovered that the population P of quail in a region is related to the percent of the region that has been paved with roads and parking lots,
According to the formula $$P = k \ln \left( \frac { 30 } { x + 4 } \right) , 0 \leq x \leq 26$$ where x is the percent of the region that has been paved. For a particular rural region, $P = 1600 \text { when } x = 0$ Predict
What the quail population will be in this region when it becomes 6% paved. Round your answer to the nearest
Whole number.

Accepted Answer

A)  883 quail 
B)  925 quail 
C)  0 quail 
D)  872 quail 
A)  883 quail 
B)  925 quail 
C)  0 quail 
D)  872 quail

Question 20

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth.
-$5 ^ { 2 x } = 2 ^ { x + 1 }$

Accepted Answer

A)  {0.76} 
B)  {1.43} 
C)  {0.43} 
D)  {0.27} 
A)  {0.76} 
B)  {1.43} 
C)  {0.43} 
D)  {0.27}

Use the definition of inverses to determine whether f and g are inverses. - $f ( x ) = 7 x - 7 , \quad g ( x ) = \frac { 1 } { 7 } x + 1$

A certain radioactive isotope has a half-life of approximately 1500 years. How many years to the nearest year would be required for a given amount of this isotope to decay to 70% of that amount?

Given $f ( x ) = \log _ { 2 } x$ , evaluate the following. (a) $\mathrm { f } \left( 2 ^ { 5 } \right)$ (b) $f \left( 2 \log _ { 2 } 2 \right)$ (c) $\mathrm { f } \left( 2 ^ { 2 } \log _ { 2 } 2 \right)$

Solve the following graphically. If necessary, round answers to the nearest thousandth. - $\log x ^ { 3 } = ( \log x ) ^ { 3 }$

Write an equivalent expression in exponential form. - $\log_{ ( x - 6 )} 18 = 1$

Find the future value. -$3169 invested for 4 years at 3% compounded annually

Round your answer to the nearest tenth, when appropriate. Use the formula $\mathrm { pH } = - \log \left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right]$ , as needed. -Find the pH if $\left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right] = 7.9 \times 10 ^ { - 3 }$ .

Round to the nearest thousandth. - $4 \mathrm { e } ^ { ( 3 \mathrm { x } + 2 ) } = 2$

Use the alphabet coding method $( A = 1 , B = 2 , C = 3 , \ldots Z = 26 )$ to solve the problem. -The function defined by $f ( x ) = x + 2$ was used to encode a message as: 15 3 22 10 9 7 16 11 23 21 Find the inverse function and determine the message.

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - $f ( x ) = - \sqrt { x ^ { 2 } - 25 } , x \geq 5$

Determine whether the statement is true or false. -If a function f has an inverse and f(1) $f ( 1 ) = - 3 , \text { then } f ^ { - 1 } ( - 3 ) = 1$

Solve the equation. - $\left( \frac { 6 } { 7 } \right) ^ { x } = \frac { 2401 } { 1296 }$

For the function as defined that is one-to-one, graph f and $\mathrm { f } ^ { - 1 }$ on the same axes. - $f(x)=2 x$ $For the function as defined that is one-to-one, graph f and \mathrm { f } ^ { - 1 } on the same axes. - f(x)=2 x$

Determine whether the statement is true or false. -If a function f has an inverse function, then f is one-to-one.

Solve the equation. -

Find the value. Give an approximation to four decimal places. -ln 0.984

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth. - $5 ^ { 2 x } = 2 ^ { x + 1 }$

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomials and Rational Functions

Systems and Matrices

Arithmetic Sequence: Common Difference and First n Terms

Filters

Exam 5: Inverse, Exponential, and Logarithmic Functions

Use the definition of inverses to determine whether f and g are inverses. - f(x)=7x−7,g(x)=17x+1f ( x ) = 7 x - 7 , \quad g ( x ) = \frac { 1 } { 7 } x + 1f(x)=7x−7,g(x)=71​x+1

A certain radioactive isotope has a half-life of approximately 1500 years. How many years to the nearest year would be required for a given amount of this isotope to decay to 70% of that amount?

Given f(x)=log⁡2xf ( x ) = \log _ { 2 } xf(x)=log2​x , evaluate the following. (a) f(25)\mathrm { f } \left( 2 ^ { 5 } \right)f(25) (b) f(2log⁡22)f \left( 2 \log _ { 2 } 2 \right)f(2log2​2) (c) f(22log⁡22)\mathrm { f } \left( 2 ^ { 2 } \log _ { 2 } 2 \right)f(22log2​2)

Solve the following graphically. If necessary, round answers to the nearest thousandth. - log⁡x3=(log⁡x)3\log x ^ { 3 } = ( \log x ) ^ { 3 }logx3=(logx)3

Write an equivalent expression in exponential form. - log⁡(x−6)18=1\log_{ ( x - 6 )} 18 = 1log(x−6)​18=1

Find the future value. -$3169 invested for 4 years at 3% compounded annually

Round to the nearest thousandth. - 4e(3x+2)=24 \mathrm { e } ^ { ( 3 \mathrm { x } + 2 ) } = 24e(3x+2)=2

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - f(x)=−x2−25,x≥5f ( x ) = - \sqrt { x ^ { 2 } - 25 } , x \geq 5f(x)=−x2−25​,x≥5

Determine whether the statement is true or false. -If a function f has an inverse and f(1) f(1)=−3, then f−1(−3)=1f ( 1 ) = - 3 , \text { then } f ^ { - 1 } ( - 3 ) = 1f(1)=−3, then f−1(−3)=1

Solve the equation. - (67)x=24011296\left( \frac { 6 } { 7 } \right) ^ { x } = \frac { 2401 } { 1296 }(76​)x=12962401​

For the function as defined that is one-to-one, graph f and f−1\mathrm { f } ^ { - 1 }f−1 on the same axes. - f(x)=2xf(x)=2 xf(x)=2x

Determine whether the statement is true or false. -If a function f has an inverse function, then f is one-to-one.

Solve the equation. -

Find the value. Give an approximation to four decimal places. -ln 0.984

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth. - 52x=2x+15 ^ { 2 x } = 2 ^ { x + 1 }52x=2x+1

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomials and Rational Functions

Systems and Matrices

Arithmetic Sequence: Common Difference and First n Terms

Filters

Use the definition of inverses to determine whether f and g are inverses. - $f ( x ) = 7 x - 7 , \quad g ( x ) = \frac { 1 } { 7 } x + 1$

Given $f ( x ) = \log _ { 2 } x$ , evaluate the following. (a) $\mathrm { f } \left( 2 ^ { 5 } \right)$ (b) $f \left( 2 \log _ { 2 } 2 \right)$ (c) $\mathrm { f } \left( 2 ^ { 2 } \log _ { 2 } 2 \right)$

Solve the following graphically. If necessary, round answers to the nearest thousandth. - $\log x ^ { 3 } = ( \log x ) ^ { 3 }$

Write an equivalent expression in exponential form. - $\log_{ ( x - 6 )} 18 = 1$

Round to the nearest thousandth. - $4 \mathrm { e } ^ { ( 3 \mathrm { x } + 2 ) } = 2$

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - $f ( x ) = - \sqrt { x ^ { 2 } - 25 } , x \geq 5$

Determine whether the statement is true or false. -If a function f has an inverse and f(1) $f ( 1 ) = - 3 , \text { then } f ^ { - 1 } ( - 3 ) = 1$

Solve the equation. - $\left( \frac { 6 } { 7 } \right) ^ { x } = \frac { 2401 } { 1296 }$

For the function as defined that is one-to-one, graph f and $\mathrm { f } ^ { - 1 }$ on the same axes. - $f(x)=2 x$ $For the function as defined that is one-to-one, graph f and \mathrm { f } ^ { - 1 } on the same axes. - f(x)=2 x$

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth. - $5 ^ { 2 x } = 2 ^ { x + 1 }$