Question 1

Determine whether the function has an inverse function.If it does, find the inverse function. $$f ( x ) = x ^ { 2 } + 5$$

Accepted Answer

A)  No inverse function exists. 
B)  $f ^ { - 1 } ( x ) = \sqrt { x } + 5 , x \geq 0$ 
C)  $f ^ { - 1 } ( x ) = \sqrt { x } - 5$ 
D)  $f ^ { - 1 } ( x ) = \sqrt { x + 5 } , x \geq - 6$ 
E)  $f ^ { - 1 } ( x ) = \sqrt { x - 5 }$ 
A)  No inverse function exists. 
B)  $f ^ { - 1 } ( x ) = \sqrt { x } + 5 , x \geq 0$ 
C)  $f ^ { - 1 } ( x ) = \sqrt { x } - 5$ 
D)  $f ^ { - 1 } ( x ) = \sqrt { x + 5 } , x \geq - 6$ 
E)  $f ^ { - 1 } ( x ) = \sqrt { x - 5 }$

Question 2

Find the domain of the function. $$g ( w ) = \frac { - 7 w } { w - 5 }$$

Accepted Answer

A)  all real numbers $w \neq 5$ 
B)  all real numbers $w \neq 5 , w \neq 0$ 
C)  all real numbers 
D)  $w = 5 , w = 0$ 
E)  $w = 5$ 
A)  all real numbers $w \neq 5$ 
B)  all real numbers $w \neq 5 , w \neq 0$ 
C)  all real numbers 
D)  $w = 5 , w = 0$ 
E)  $w = 5$

Question 3

Find $f \circ g$.
$$f ( x ) = x + 2 \quad g ( x ) = \frac { 5 } { x ^ { 2 } - 4 }$$

Accepted Answer

A)  $( f \circ g ) ( x ) = \frac { 5 } { x ^ { 2 } }$ 
B)  $( f \circ g ) ( x ) = \frac { 5 } { x ^ { 2 } + 4 x }$ 
C) 
$( f \circ g ) ( x ) = \frac { 2 x ^ { 2 } + 3 } { x ^ { 2 } - 4 }$ 
D)  $( f \circ g ) ( x ) = \frac { 7 } { x ^ { 2 } - 4 }$ 
E) 
$( f \circ g ) ( x ) = \frac { 2 x ^ { 2 } - 3 } { x ^ { 2 } - 4 }$ 
A)  $( f \circ g ) ( x ) = \frac { 5 } { x ^ { 2 } }$ 
B)  $( f \circ g ) ( x ) = \frac { 5 } { x ^ { 2 } + 4 x }$ 
C) 
$( f \circ g ) ( x ) = \frac { 2 x ^ { 2 } + 3 } { x ^ { 2 } - 4 }$ 
D)  $( f \circ g ) ( x ) = \frac { 7 } { x ^ { 2 } - 4 }$ 
E) 
$( f \circ g ) ( x ) = \frac { 2 x ^ { 2 } - 3 } { x ^ { 2 } - 4 }$

Question 4

Which equation does not represent $y$ as a function of $x$ ?

Accepted Answer

A)  $x = - 9 y + 2$ 
B)  $x = - 1$ 
C)  $y = 7 x - 9$ 
D)  $y = \left| 6 - x ^ { 2 } \right|$ 
E)  $y = \sqrt { - 9 + 6 x }$ 
A)  $x = - 9 y + 2$ 
B)  $x = - 1$ 
C)  $y = 7 x - 9$ 
D)  $y = \left| 6 - x ^ { 2 } \right|$ 
E)  $y = \sqrt { - 9 + 6 x }$

Question 5

Determine the domain of $g ( x ) = \frac { 1 } { x ^ { 2 } - 49 }$.

Accepted Answer

A)  $[ - 7,7 ]$ 
B)  $( - 7,0 ] \cup [ 0,7 )$ 
C)  $( - \infty , - 7 ) \cup ( - 7,7 ) \cup ( 7 , \infty )$ 
D)  $( - \infty , - 7 ] \cup [ 7 , \infty )$ 
E)  $( - \infty , \infty )$ 
A)  $[ - 7,7 ]$ 
B)  $( - 7,0 ] \cup [ 0,7 )$ 
C)  $( - \infty , - 7 ) \cup ( - 7,7 ) \cup ( 7 , \infty )$ 
D)  $( - \infty , - 7 ] \cup [ 7 , \infty )$ 
E)  $( - \infty , \infty )$

Question 6

Determine an equation that may represented by the graph shown below.

Accepted Answer

A)  $f ( x ) = | x + 1 | + 1$ 
B)  $f ( x ) = | x + 1 | - 1$ 
C)  $\quad f ( x ) = - | x - 1 | + 1$ 
D)  $f ( x ) = | x - 1 | + 1$ 
E)  $f ( x ) = | x - 1 | - 1$ 
A)  $f ( x ) = | x + 1 | + 1$ 
B)  $f ( x ) = | x + 1 | - 1$ 
C)  $\quad f ( x ) = - | x - 1 | + 1$ 
D)  $f ( x ) = | x - 1 | + 1$ 
E)  $f ( x ) = | x - 1 | - 1$

Question 7

Find the slope-intercept form of the line passing through the points. $( - 4 , - 2 ) , ( - 1,7 )$

Accepted Answer

A)  $y = 3 x + 2$ 
B)  $y = 3 x + 10$ 
C)  $y = \frac { 1 } { 3 } x - \frac { 2 } { 3 }$ 
D)  $y = - \frac { 1 } { 3 } x - \frac { 10 } { 3 }$ 
E)  $y = - 3 x - 14$ 
A)  $y = 3 x + 2$ 
B)  $y = 3 x + 10$ 
C)  $y = \frac { 1 } { 3 } x - \frac { 2 } { 3 }$ 
D)  $y = - \frac { 1 } { 3 } x - \frac { 10 } { 3 }$ 
E)  $y = - 3 x - 14$

Question 8

Determine an equation that may represented by the graph shown below.

Accepted Answer

A)  $f ( x ) = ( x - 1 ) ^ { 3 } + 2$ 
B)  $f ( x ) = - ( x - 1 ) ^ { 3 } + 2$ 
C)  $f ( x ) = - ( x - 1 ) ^ { 3 } - 2$ 
D)  $f ( x ) = - ( x + 1 ) ^ { 3 } - 2$ 
E)  $f ( x ) = - ( x + 1 ) ^ { 3 } + 2$ 
A)  $f ( x ) = ( x - 1 ) ^ { 3 } + 2$ 
B)  $f ( x ) = - ( x - 1 ) ^ { 3 } + 2$ 
C)  $f ( x ) = - ( x - 1 ) ^ { 3 } - 2$ 
D)  $f ( x ) = - ( x + 1 ) ^ { 3 } - 2$ 
E)  $f ( x ) = - ( x + 1 ) ^ { 3 } + 2$

Question 9

Decide whether the two functions shown in the graph below appear to be inverse functions of each other.

Accepted Answer

A)  no 
B)  yes 
C)  not enough information 
A)  no 
B)  yes 
C)  not enough information

Question 10

Find the value(s) of $x$ for which $f ( x ) = g ( x )$.
$$f ( x ) = x ^ { 2 } - 13 x + 5 \quad g ( x ) = - 9 x + 2$$

Accepted Answer

A)  $5,18 , \frac { 2 } { 9 }$ 
B)  $5 , - 13 , \frac { 2 } { 9 }$ 
C)  3,1 
D)  $- 3 , - 1$ 
E)  $8 , \frac { 2 } { 9 }$ 
A)  $5,18 , \frac { 2 } { 9 }$ 
B)  $5 , - 13 , \frac { 2 } { 9 }$ 
C)  3,1 
D)  $- 3 , - 1$ 
E)  $8 , \frac { 2 } { 9 }$

Question 11

Use the functions $f ( x ) = x + 4$ and $g ( x ) = 5 x - 7$ to find $( g \circ f ) ^ { - 1 }$.

Accepted Answer

A)  $( g \circ f ) ^ { - 1 } = \frac { 5 x + 11 } { 4 }$ 
B)  $( g \circ f ) ^ { - 1 } = 5 x - 42$ 
C)  $( g \circ f ) ^ { - 1 } = \frac { x - 13 } { 5 }$ 
D)  $( g \circ f ) ^ { - 1 } = \frac { - 7 x - 7 } { 5 }$ 
E)  $( g \circ f ) ^ { - 1 } = 5 x + 13$ 
A)  $( g \circ f ) ^ { - 1 } = \frac { 5 x + 11 } { 4 }$ 
B)  $( g \circ f ) ^ { - 1 } = 5 x - 42$ 
C)  $( g \circ f ) ^ { - 1 } = \frac { x - 13 } { 5 }$ 
D)  $( g \circ f ) ^ { - 1 } = \frac { - 7 x - 7 } { 5 }$ 
E)  $( g \circ f ) ^ { - 1 } = 5 x + 13$

Question 12

Determine a piecewise-defined function for the graph shown below.

Accepted Answer

A) 
$$f ( x ) = \left\{ \begin{array} { l l } 
| x | , & x \leq 1 \
- ( x - 1 ) ^ { 2 } + 1 , & x > 1
\end{array} \right.$$ 
B) 
$$f ( x ) = \left\{ \begin{array} { l l } 
| x | , & x \leq 0 \
- ( x - 1 ) ^ { 2 } + 1 , & x \leq 0
\end{array} \right.$$ 
C) 
$$f ( x ) = \left\{ \begin{array} { l l } 
| x | , & x \geq 1 \
- x ^ { 2 } , & x \leq 1
\end{array} \right.$$ 
D) 
$$f ( x ) = \left\{ \begin{array} { l l } 
| x | , & x \geq 0 \
- x ^ { 2 } , & x \leq 1
\end{array} \right.$$ 
E) 
$$f ( x ) = \left\{ \begin{array} { l l } 
| x | , & x \leq 1 \
- ( x - 1 ) ^ { 2 } , & x > 1
\end{array} \right.$$ 
A) 
$$f ( x ) = \left\{ \begin{array} { l l } 
| x | , & x \leq 1 \
- ( x - 1 ) ^ { 2 } + 1 , & x > 1
\end{array} \right.$$ 
B) 
$$f ( x ) = \left\{ \begin{array} { l l } 
| x | , & x \leq 0 \
- ( x - 1 ) ^ { 2 } + 1 , & x \leq 0
\end{array} \right.$$ 
C) 
$$f ( x ) = \left\{ \begin{array} { l l } 
| x | , & x \geq 1 \
- x ^ { 2 } , & x \leq 1
\end{array} \right.$$ 
D) 
$$f ( x ) = \left\{ \begin{array} { l l } 
| x | , & x \geq 0 \
- x ^ { 2 } , & x \leq 1
\end{array} \right.$$ 
E) 
$$f ( x ) = \left\{ \begin{array} { l l } 
| x | , & x \leq 1 \
- ( x - 1 ) ^ { 2 } , & x > 1
\end{array} \right.$$

Question 13

Determine the domain and range of the inverse function $f ^ { - 1 }$ of the following function $f$ $f ( x ) = - | x + 7 | - 1$, where $x > - 7$

Accepted Answer

A)  Domain: $[ - 7 , \infty )$; Range: $[ - 1 , \infty )$ 
B)  Domain: $( - \infty , - 1 ]$; Range: $[ - 7 , \infty )$ 
C)  Domain: $[ - 7 , - 1 ]$; Range: $[ - 7 , \infty )$ 
D)  Domain: $( - \infty , - 7 ]$; Range: $[ 1 , \infty )$ 
E)  Domain: $( - \infty , \infty )$; Range: $( - \infty , \infty )$ 
A)  Domain: $[ - 7 , \infty )$; Range: $[ - 1 , \infty )$ 
B)  Domain: $( - \infty , - 1 ]$; Range: $[ - 7 , \infty )$ 
C)  Domain: $[ - 7 , - 1 ]$; Range: $[ - 7 , \infty )$ 
D)  Domain: $( - \infty , - 7 ]$; Range: $[ 1 , \infty )$ 
E)  Domain: $( - \infty , \infty )$; Range: $( - \infty , \infty )$

Question 14

Use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant. $f ( x ) = - x ^ { 3 } + 3 x + 1$

Accepted Answer

A) 
increasing on $( - \infty , - 1 )$
decreasing on $( - 1,1 )$
increasing on $( 1 , \infty )$ 
B) 
decreasing on $( - \infty , 0 )$
increasing on $( 0 , \infty )$ 
C) 
decreasing on $( - \infty , \infty )$ 
D) 
increasing on $( - \infty , \infty )$ 
E) 
decreasing on $( - \infty , - 1 )$
increasing on $( - 1,1 )$
decreasing on $( 1 , \infty )$ 
A) 
increasing on $( - \infty , - 1 )$
decreasing on $( - 1,1 )$
increasing on $( 1 , \infty )$ 
B) 
decreasing on $( - \infty , 0 )$
increasing on $( 0 , \infty )$ 
C) 
decreasing on $( - \infty , \infty )$ 
D) 
increasing on $( - \infty , \infty )$ 
E) 
decreasing on $( - \infty , - 1 )$
increasing on $( - 1,1 )$
decreasing on $( 1 , \infty )$

Question 15

Find the domain of the function. $$g ( w ) = \frac { 4 w } { w + 9 }$$

Accepted Answer

A)  all real numbers $w \neq - 9$ 
B)  all real numbers $w \neq - 9 , w \neq 0$ 
C)  all real numbers 
D)  $w = - 9 , w = 0$ 
E)  $w = - 9$ 
A)  all real numbers $w \neq - 9$ 
B)  all real numbers $w \neq - 9 , w \neq 0$ 
C)  all real numbers 
D)  $w = - 9 , w = 0$ 
E)  $w = - 9$

Question 16

Show algebraically that the functions $f$ and $g$ shown below are inverse functions.
$$f ( x ) = \frac { 2 } { 2 + x } , x \geq 0 , \quad g ( x ) = \frac { 2 - 2 x } { x } , 0 < x \leq 1$$

Accepted Answer

A) 
$\begin{array}{l}
\begin{array}{lll}
f(g(x)) &=\frac{2}{2+\left(\frac{2-2 x}{x}\right)} \
&=\frac{2}{2+\left(\frac{1}{x}\right)} \
&=\frac{1}{\left(\frac{1}{x}\right)} \
&=1 \cdot \frac{x}{1} \
&=x
\end{array}
\begin{array}{lll}
g(f(x)) &=\frac{2-2\left(\frac{2}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{0-\left(\frac{2}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{\frac{-2}{2+x}}{\left(\frac{2}{2+x}\right)} \
&=\left(\frac{-2}{2+x}\right)\left(\frac{2+x}{2}\right) \
&=\frac{2 x+2}{2+x} \
&=x
\end{array}
\end{array}$ 
B) 
$\begin{array}{l}
\begin{array}{lll}
f(g(x)) &=\frac{2}{2+\left(\frac{2-2 x}{x}\right)} \
&=\frac{1}{1+\frac{2-2 x}{x}} \
&=\frac{1}{\left(\frac{0}{x}\right)} \
&=x
\end{array}
\begin{array}{lll}
g(f(x)) &=\frac{2-2\left(\frac{2}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{2-\left(\frac{4}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{\frac{4+2 x-4}{2+x}}{\left(\frac{2}{2+x}\right)} \
&=\frac{2 x}{\left(\frac{x}{2+x}\right).} \
&=\frac{2 x}{x} \
&= x
\end{array}
\end{array}$ 
C) 
$\begin{array}{l}
\begin{array}{lll}
f(g(x)) &=\frac{2}{2+\left(\frac{2-2 x}{x}\right)} \
&=\frac{4}{\frac{2-2 x}{x}} \
&=\frac{2}{\left(\frac{2 x}{x}\right)} \
&=2 \cdot \frac{x}{2} \
&=x
\end{array}
\begin{array}{lll}
g(f(x)) &=\frac{2-2\left(\frac{2}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{\left(\frac{2}{2+2 x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\left(\frac{2}{2+2 x}\right)\left(\frac{2+x}{2}\right) \
&=\frac{2+x}{2+2 x} \
&=\frac{x}{2 x} \
&=x
\end{array}
\end{array}$ 
D) 
$\begin{array}{l}
\begin{array}{lll}
f(g(x)) &=\frac{2}{2+\left(\frac{2-2 x}{x}\right)} \
&=\frac{2}{\frac{2 x+2-2 x}{x}} \
&=\frac{2-2 x}{\left(\frac{2}{x}\right)} \
&=\frac{1-x}{1} \
&=x
\end{array}
\begin{array}{lll}
g(f(x)) &=\frac{2-2\left(\frac{2}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{\left(\frac{4}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\left(\frac{4}{2+x}\right)\left(\frac{2+x}{2}\right) \
&=\frac{2(2+x)}{2+x} \
&=\frac{2 x}{2}\
&=x
\end{array}
\end{array}$ 
E) 
$\begin{array}{l}
\begin{array}{lll}
f(g(x)) &=\frac{2}{2+\left(\frac{2-2 x}{x}\right)} \
&=\frac{2}{\frac{2 x+2-2 x}{x}} \
&=\frac{2}{\left(\frac{2}{x}\right)} \
&=2 \cdot \frac{x}{2} \
&=x
\end{array}
\begin{array}{lll}
g(f(x)) &=\frac{2-2\left(\frac{2}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{2-\left(\frac{4}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{\frac{4+2 x-4}{2+x}}{\left(\frac{2}{2+x}\right)} \
&=\frac{\frac{2 x}{2+x}}{\left(\frac{2}{2+x}\right)} \
&=\frac{2 x}{2}
\end{array}
\end{array}$ 
A) 
$\begin{array}{l}
\begin{array}{lll}
f(g(x)) &=\frac{2}{2+\left(\frac{2-2 x}{x}\right)} \
&=\frac{2}{2+\left(\frac{1}{x}\right)} \
&=\frac{1}{\left(\frac{1}{x}\right)} \
&=1 \cdot \frac{x}{1} \
&=x
\end{array}
\begin{array}{lll}
g(f(x)) &=\frac{2-2\left(\frac{2}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{0-\left(\frac{2}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{\frac{-2}{2+x}}{\left(\frac{2}{2+x}\right)} \
&=\left(\frac{-2}{2+x}\right)\left(\frac{2+x}{2}\right) \
&=\frac{2 x+2}{2+x} \
&=x
\end{array}
\end{array}$ 
B) 
$\begin{array}{l}
\begin{array}{lll}
f(g(x)) &=\frac{2}{2+\left(\frac{2-2 x}{x}\right)} \
&=\frac{1}{1+\frac{2-2 x}{x}} \
&=\frac{1}{\left(\frac{0}{x}\right)} \
&=x
\end{array}
\begin{array}{lll}
g(f(x)) &=\frac{2-2\left(\frac{2}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{2-\left(\frac{4}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{\frac{4+2 x-4}{2+x}}{\left(\frac{2}{2+x}\right)} \
&=\frac{2 x}{\left(\frac{x}{2+x}\right).} \
&=\frac{2 x}{x} \
&= x
\end{array}
\end{array}$ 
C) 
$\begin{array}{l}
\begin{array}{lll}
f(g(x)) &=\frac{2}{2+\left(\frac{2-2 x}{x}\right)} \
&=\frac{4}{\frac{2-2 x}{x}} \
&=\frac{2}{\left(\frac{2 x}{x}\right)} \
&=2 \cdot \frac{x}{2} \
&=x
\end{array}
\begin{array}{lll}
g(f(x)) &=\frac{2-2\left(\frac{2}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{\left(\frac{2}{2+2 x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\left(\frac{2}{2+2 x}\right)\left(\frac{2+x}{2}\right) \
&=\frac{2+x}{2+2 x} \
&=\frac{x}{2 x} \
&=x
\end{array}
\end{array}$ 
D) 
$\begin{array}{l}
\begin{array}{lll}
f(g(x)) &=\frac{2}{2+\left(\frac{2-2 x}{x}\right)} \
&=\frac{2}{\frac{2 x+2-2 x}{x}} \
&=\frac{2-2 x}{\left(\frac{2}{x}\right)} \
&=\frac{1-x}{1} \
&=x
\end{array}
\begin{array}{lll}
g(f(x)) &=\frac{2-2\left(\frac{2}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{\left(\frac{4}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\left(\frac{4}{2+x}\right)\left(\frac{2+x}{2}\right) \
&=\frac{2(2+x)}{2+x} \
&=\frac{2 x}{2}\
&=x
\end{array}
\end{array}$ 
E) 
$\begin{array}{l}
\begin{array}{lll}
f(g(x)) &=\frac{2}{2+\left(\frac{2-2 x}{x}\right)} \
&=\frac{2}{\frac{2 x+2-2 x}{x}} \
&=\frac{2}{\left(\frac{2}{x}\right)} \
&=2 \cdot \frac{x}{2} \
&=x
\end{array}
\begin{array}{lll}
g(f(x)) &=\frac{2-2\left(\frac{2}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{2-\left(\frac{4}{2+x}\right)}{\left(\frac{2}{2+x}\right)} \
&=\frac{\frac{4+2 x-4}{2+x}}{\left(\frac{2}{2+x}\right)} \
&=\frac{\frac{2 x}{2+x}}{\left(\frac{2}{2+x}\right)} \
&=\frac{2 x}{2}
\end{array}
\end{array}$

Question 17

Use the graph of $f$ to sketch the graph of the function indicated below.

Accepted Answer

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

Question 18

The scatter plots of different data are shown below. Determine whether there is a positive correlation, negative correlation, or no discernible correlation between the variables.

Accepted Answer

A)  positive correlation 
B)  negative correlation 
C)  no discernible correlation 
A)  positive correlation 
B)  negative correlation 
C)  no discernible correlation

Question 19

Use the graphs of $y = f ( x )$ and $y = g ( x )$ to evaluate $\left( g ^ { - 1 } \circ f ^ { - 1 } \right) ( - 4 )$

Accepted Answer

A)  4 
B)  $1.3$ 
C)  0 
D)  $- 2$ 
E)  $- 2.5$ 
A)  4 
B)  $1.3$ 
C)  0 
D)  $- 2$ 
E)  $- 2.5$

Question 20

Hooke's Law states that the force $F$ required to compress or stretch a spring (within its elastic limits) is proportional to the distance $d$ that the spring is compressed or stretched from its original length. That is, $F = k d$, where $k$ is the measure of the stiffness of the spring and is called the spring constant. The table below shows the elongation $d$ in centimeters of a spring when a force of $F$ kilograms is applied.
$\begin{array}{|c|c|}
\hline \text { Force, } \boldsymbol{F} & \text { Elongation, } \boldsymbol{d} \
\hline 20 & 1.4 \
\hline 40 & 2.5 \
\hline 60 & 4.0 \
\hline 80 & 5.3 \
\hline 100 & 6.6 \
\hline
\end{array}$

Find the equation of the line that seems to best fit the data. Use the model to estimate the elongation of the spring when a force of 55 kilograms is applied. Round your answer to one decimal place.

Accepted Answer

A)  $7.2$ centimeters 
B)  $5.4$ centimeters 
C)  $1.8$ centimeters 
D)  $3.6$ centimeters 
E)  $2.7$ centimeters 
A)  $7.2$ centimeters 
B)  $5.4$ centimeters 
C)  $1.8$ centimeters 
D)  $3.6$ centimeters 
E)  $2.7$ centimeters

Determine whether the function has an inverse function.If it does, find the inverse function. $f ( x ) = x ^ { 2 } + 5$

Find the domain of the function. $g ( w ) = \frac { - 7 w } { w - 5 }$

Find $f \circ g$ . $f ( x ) = x + 2 \quad g ( x ) = \frac { 5 } { x ^ { 2 } - 4 }$

Which equation does not represent $y$ as a function of $x$ ?

Determine the domain of $g ( x ) = \frac { 1 } { x ^ { 2 } - 49 }$ .

Determine an equation that may represented by the graph shown below.

Find the slope-intercept form of the line passing through the points. $( - 4 , - 2 ) , ( - 1,7 )$

Determine an equation that may represented by the graph shown below.

Decide whether the two functions shown in the graph below appear to be inverse functions of each other.

Find the value(s) of $x$ for which $f ( x ) = g ( x )$ . $f ( x ) = x ^ { 2 } - 13 x + 5 \quad g ( x ) = - 9 x + 2$

Use the functions $f ( x ) = x + 4$ and $g ( x ) = 5 x - 7$ to find $( g \circ f ) ^ { - 1 }$ .

Determine a piecewise-defined function for the graph shown below.

Determine the domain and range of the inverse function $f ^ { - 1 }$ of the following function $f$ $f ( x ) = - | x + 7 | - 1$ , where $x > - 7$

Use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant. $f ( x ) = - x ^ { 3 } + 3 x + 1$

Find the domain of the function. $g ( w ) = \frac { 4 w } { w + 9 }$

Show algebraically that the functions $f$ and $g$ shown below are inverse functions. $f ( x ) = \frac { 2 } { 2 + x } , x \geq 0 , \quad g ( x ) = \frac { 2 - 2 x } { x } , 0 < x \leq 1$

Use the graph of $f$ to sketch the graph of the function indicated below.

The scatter plots of different data are shown below. Determine whether there is a positive correlation, negative correlation, or no discernible correlation between the variables.

Use the graphs of $y = f ( x )$ and $y = g ( x )$ to evaluate $\left( g ^ { - 1 } \circ f ^ { - 1 } \right) ( - 4 )$ $Use the graphs of y = f ( x ) and y = g ( x ) to evaluate \left( g ^ { - 1 } \circ f ^ { - 1 } \right) ( - 4 )$

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

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Exam 1: Functions and Their Graphs

Determine whether the function has an inverse function.If it does, find the inverse function. f(x)=x2+5f ( x ) = x ^ { 2 } + 5f(x)=x2+5

Find the domain of the function. g(w)=−7ww−5g ( w ) = \frac { - 7 w } { w - 5 }g(w)=w−5−7w​

Find f∘gf \circ gf∘g . f(x)=x+2g(x)=5x2−4f ( x ) = x + 2 \quad g ( x ) = \frac { 5 } { x ^ { 2 } - 4 }f(x)=x+2g(x)=x2−45​

Which equation does not represent yyy as a function of xxx ?

Determine the domain of g(x)=1x2−49g ( x ) = \frac { 1 } { x ^ { 2 } - 49 }g(x)=x2−491​ .

Determine an equation that may represented by the graph shown below.

Find the slope-intercept form of the line passing through the points. (−4,−2),(−1,7)( - 4 , - 2 ) , ( - 1,7 )(−4,−2),(−1,7)

Determine an equation that may represented by the graph shown below.

Decide whether the two functions shown in the graph below appear to be inverse functions of each other.

Find the value(s) of xxx for which f(x)=g(x)f ( x ) = g ( x )f(x)=g(x) . f(x)=x2−13x+5g(x)=−9x+2f ( x ) = x ^ { 2 } - 13 x + 5 \quad g ( x ) = - 9 x + 2f(x)=x2−13x+5g(x)=−9x+2

Use the functions f(x)=x+4f ( x ) = x + 4f(x)=x+4 and g(x)=5x−7g ( x ) = 5 x - 7g(x)=5x−7 to find (g∘f)−1( g \circ f ) ^ { - 1 }(g∘f)−1 .

Determine a piecewise-defined function for the graph shown below.

Determine the domain and range of the inverse function f−1f ^ { - 1 }f−1 of the following function fff f(x)=−∣x+7∣−1f ( x ) = - | x + 7 | - 1f(x)=−∣x+7∣−1 , where x>−7x > - 7x>−7

Use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant. f(x)=−x3+3x+1f ( x ) = - x ^ { 3 } + 3 x + 1f(x)=−x3+3x+1

Find the domain of the function. g(w)=4ww+9g ( w ) = \frac { 4 w } { w + 9 }g(w)=w+94w​

Show algebraically that the functions fff and ggg shown below are inverse functions. f(x)=22+x,x≥0,g(x)=2−2xx,0<x≤1f ( x ) = \frac { 2 } { 2 + x } , x \geq 0 , \quad g ( x ) = \frac { 2 - 2 x } { x } , 0 < x \leq 1f(x)=2+x2​,x≥0,g(x)=x2−2x​,0<x≤1

Use the graph of fff to sketch the graph of the function indicated below.

The scatter plots of different data are shown below. Determine whether there is a positive correlation, negative correlation, or no discernible correlation between the variables.

Use the graphs of y=f(x)y = f ( x )y=f(x) and y=g(x)y = g ( x )y=g(x) to evaluate (g−1∘f−1)(−4)\left( g ^ { - 1 } \circ f ^ { - 1 } \right) ( - 4 )(g−1∘f−1)(−4)

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Calculus Practice Problems

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Determine whether the function has an inverse function.If it does, find the inverse function. $f ( x ) = x ^ { 2 } + 5$

Find the domain of the function. $g ( w ) = \frac { - 7 w } { w - 5 }$

Find $f \circ g$ . $f ( x ) = x + 2 \quad g ( x ) = \frac { 5 } { x ^ { 2 } - 4 }$

Which equation does not represent $y$ as a function of $x$ ?

Determine the domain of $g ( x ) = \frac { 1 } { x ^ { 2 } - 49 }$ .

Find the slope-intercept form of the line passing through the points. $( - 4 , - 2 ) , ( - 1,7 )$

Find the value(s) of $x$ for which $f ( x ) = g ( x )$ . $f ( x ) = x ^ { 2 } - 13 x + 5 \quad g ( x ) = - 9 x + 2$

Use the functions $f ( x ) = x + 4$ and $g ( x ) = 5 x - 7$ to find $( g \circ f ) ^ { - 1 }$ .

Determine the domain and range of the inverse function $f ^ { - 1 }$ of the following function $f$ $f ( x ) = - | x + 7 | - 1$ , where $x > - 7$

Use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant. $f ( x ) = - x ^ { 3 } + 3 x + 1$

Find the domain of the function. $g ( w ) = \frac { 4 w } { w + 9 }$

Show algebraically that the functions $f$ and $g$ shown below are inverse functions. $f ( x ) = \frac { 2 } { 2 + x } , x \geq 0 , \quad g ( x ) = \frac { 2 - 2 x } { x } , 0 < x \leq 1$

Use the graph of $f$ to sketch the graph of the function indicated below.

Use the graphs of $y = f ( x )$ and $y = g ( x )$ to evaluate $\left( g ^ { - 1 } \circ f ^ { - 1 } \right) ( - 4 )$ $Use the graphs of y = f ( x ) and y = g ( x ) to evaluate \left( g ^ { - 1 } \circ f ^ { - 1 } \right) ( - 4 )$