Question 1

Restrict the domain of the following function $f$ so that the function is one-to-one and has an inverse function.
$f ( x ) = - | x - 4 | + 2$

Accepted Answer

A)  $[ - 4 , \infty )$ 
B)  $[ 2,4 ]$ 
C)  $[ 4 , \infty )$ 
D)  $[ - 2,4 ]$ 
E)  $( - \infty , 2 ]$

Question 2

Does the table describe a function?
$\begin{array}{|c|c|c|c|c|c|}
\hline \text { Input value } & -6 & -3 & 0 & 3 & 6 \
\hline \text { Output value } & 11 & 11 & 11 & 11 & 11 \
\hline
\end{array}$

Accepted Answer

A)  yes 
B)  no

Question 3

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

Question 4

If $f$ is an even function, determine if $g$ is even, odd, or neither. $g ( x ) = - f ( x - 2 )$

Accepted Answer

A)  even 
B)  odd 
C)  cannot be determined 
D)  neither

Question 5

Use function notation to write $g$ in terms of $f ( x ) = x ^ { 3 }$.
$$g ( x ) = - \frac { 1 } { 5 } ( x + 4 ) ^ { 3 }$$

Accepted Answer

A)  $\quad g ( x ) = - \frac { 1 } { 5 } [ f ( x ) ] ^ { 3 } + 4$ 
B)  $g ( x ) = - \frac { 1 } { 5 } [ f ( x ) + 4 ]$ 
C)  $g ( x ) = - [ f ( x ) ] ^ { 3 } + \frac { 64 } { 5 }$ 
D)  $g ( x ) = - \frac { 1 } { 5 } [ f ( x ) ] ^ { 3 } + 64$ 
E)  $\quad g ( x ) = - \frac { 1 } { 5 } [ f ( x + 4 ) ]$

Question 6

Given the following function, $h ( x )$, find two functions $f$ and $g$ such that $( f \circ g ) ( x ) = h ( x )$.
$h ( x ) = \sqrt [ 3 ] { x ^ { 2 } - 11 }$

Accepted Answer

A)  $f ( x ) = \sqrt [ 3 ] { x ^ { 2 } } , g ( x ) = - 11$ 
B)  $f ( x ) = \sqrt [ 3 ] { x ^ { 2 } } , g ( x ) = x - 11$ 
C)  $f ( x ) = \sqrt [ 3 ] { x } , g ( x ) = x - 11$ 
D)  $f ( x ) = \sqrt [ 3 ] { x - 11 } , g ( x ) = x ^ { 2 }$ 
E)  $f ( x ) = \sqrt [ 3 ] { x - 11 } , \quad g ( x ) = x + 11$

Question 7

Use function notation to write $g$ in terms of $f ( x ) = x ^ { 3 }$.
$g ( x ) = - \frac { 1 } { 2 } ( x + 9 ) ^ { 3 }$

Accepted Answer

A)  $g ( x ) = - \frac { 1 } { 2 } [ f ( x ) ] ^ { 3 } + 9$ 
B)  $g ( x ) = - \frac { 1 } { 2 } [ f ( x ) + 9 ]$ 
C)  $\quad g ( x ) = - [ f ( x ) ] ^ { 3 } + \frac { 729 } { 2 }$ 
D)  $g ( x ) = - \frac { 1 } { 2 } [ f ( x ) ] ^ { 3 } + 729$ 
E)  $\quad g ( x ) = - \frac { 1 } { 2 } [ f ( x + 9 ) ]$

Question 8

Find all real values of $x$ such that $f ( x ) = 0$.
$f ( x ) = \frac { - 3 x - 2 } { 5 }$

Accepted Answer

A)  $- \frac { 2 } { 15 }$ 
B)  $\pm \frac { 2 } { 15 }$ 
C)  $\pm \frac { 2 } { 3 }$ 
D)  $- \frac { 2 } { 3 }$ 
E)  $\frac { 2 } { 3 }$

Question 9

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)  $8 , \frac { 2 } { 9 }$

Question 10

Determine the domain of $g ( x ) = \frac { 1 } { x ^ { 2 } - 81 }$.

Accepted Answer

A)  $[ - 9,9 ]$ 
B)  $( - 9,0 ] \cup [ 0,9 )$ 
C)  $( - \infty , - 9 ) \cup ( - 9,9 ) \cup ( 9 , \infty )$ 
D)  $( - \infty , - 9 ] \cup [ 9 , \infty )$ 
E)  $( - \infty , \infty )$

Question 11

Find the domain of the function.
$f ( t ) = \sqrt { 64 - t ^ { 2 } }$

Accepted Answer

A)  $- 8 \leq t \leq 8$ 
B)  $t \leq - 8$ or $t \geq 8$ 
C)  $t \geq 0$ 
D)  $t \leq 8$ 
E)  all real numbers

Question 12

Find the value(s) of $x$ for which $f ( x ) = g ( x )$.
$f ( x ) = x ^ { 2 } - 11 x - 36$
$g ( x ) = - 7 x - 4$

Accepted Answer

A)  $- 36 , - 25 , - \frac { 4 } { 7 }$ 
B)  $- 36 , - 11 , - \frac { 4 } { 7 }$ 
C)  $8 , - 4$ 
D)  $- 8,4$ 
E)  $47 , - \frac { 4 } { 7 }$

Question 13

Find the inverse function of $f$.
$f ( x ) = x ^ { 5 } - 1$

Accepted Answer

A)  $f ^ { - 1 } ( x ) = - \sqrt [ 5 ] { x } - 1$ 
B)  $f ^ { - 1 } ( x ) = \sqrt [ 5 ] { x } - 1$ 
C)  $f ^ { - 1 } ( x ) = - \sqrt [ 5 ] { x - 1 }$ 
D)  $f ^ { - 1 } ( x ) = \sqrt [ 5 ] { x + 1 }$ 
E)  $f ^ { - 1 } ( x ) = \sqrt [ 5 ] { x } + 1$

Question 14

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

Accepted Answer

A)  $x = 2 y + 5$ 
B)  $\quad x = 6$ 
C)  $y = - 5 x - 7$ 
D)  $y = \left| 6 + 9 x ^ { 2 } \right|$ 
E)  $y = \sqrt { - 8 + 4 x }$

Question 15

Graph the function and determine the interval(s) for which $f ( x ) \geq 0$.
$f ( x ) = - x ^ { 2 } + 4 x$

Accepted Answer

A)  $( - \infty , 0 ] \cup [ 4 , \infty )$ 
B)  $[ 0,4 ]$ 
C)  $( 0,4 )$ 
D)  $( - \infty , 0 ) \cup ( 4 , \infty )$ 
E)  $\{ 4 \}$

Question 16

Use the graph of the function to find the domain and range of $f$.

Accepted Answer

A)  domain : $( - \infty , - 2 ) \cup ( - 2 , \infty )$
range: $( - \infty , - 2 ) \cup ( - 1 , \infty )$ 
B)  domain: $( - \infty , - 2 ) \cup ( - 2 , \infty )$
range: $\{ - 2 , - 1 \}$ 
C)  domain : all real numbers
range: $\{ - 2 , - 1 \}$ 
D)  domain : $( - \infty , - 2 ) \cup ( - 2 , \infty )$
range: $( - 1,1 )$ 
E)  domain: $\{ - 2 , - 1 \}$
range: all real numbers

Question 17

Compare the graph of the following function with the graph of $f ( x ) = | x |$.
$y = \left| \frac { 7 } { 9 } x \right|$

Accepted Answer

A)  vertical shift of $\frac { 7 } { 9 }$ units up 
B)  horizontal stretch of $\frac { 9 } { 7 }$ units 
C)  vertical shrink of $\frac { 7 } { 9 }$ units 
D)  horizontal shrink of $\frac { 7 } { 9 }$ units vertical shift of $\frac { 9 } { 7 }$ units 
E)  horizontal shrink of $\frac { 7 } { 9 }$ units

Question 18

Find $f \circ g$.
$f ( x ) = x + 3 \quad g ( x ) = \frac { 4 } { x ^ { 2 } - 9 }$

Accepted Answer

A)  $( f \circ g ) ( x ) = \frac { 4 } { x ^ { 2 } }$ 
B)  $( f \circ g ) ( x ) = \frac { 4 } { x ^ { 2 } + 6 x }$ 
C)  $( f \circ g ) ( x ) = \frac { 3 x ^ { 2 } + 1 } { x ^ { 2 } - 9 }$ 
D)  $( f \circ g ) ( x ) = \frac { 7 } { x ^ { 2 } - 9 }$ 
E)  $( f \circ g ) ( x ) = \frac { 3 x ^ { 2 } - 23 } { x ^ { 2 } - 9 }$

Question 19

Find the slope and $y$-intercept of the equation of the line.
$y = - 2 x - 9$

Accepted Answer

A)  slope: $- \frac { 1 } { 2 } ; y$-intercept: $- 9$ 
B)  slope: $- \frac { 1 } { 9 } ; y$-intercept: $- 2$ 
C)  slope: $- 2 ; y$-intercept: $- 9$ 
D)  slope: $- 9 ; y$-intercept: $- 2$ 
E)  slope: $- 2 ; y$-intercept: 9

Question 20

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

Accepted Answer

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

Exam 1: Functions and Their Graphs

Restrict the domain of the following function $f$ so that the function is one-to-one and has an inverse function. $f ( x ) = - | x - 4 | + 2$

Does the table describe a function? $\begin{array}{|c|c|c|c|c|c|}\hline \text { Input value } & -6 & -3 & 0 & 3 & 6 \\\hline \text { Output value } & 11 & 11 & 11 & 11 & 11 \\\hline\end{array}$

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.

If $f$ is an even function, determine if $g$ is even, odd, or neither. $g ( x ) = - f ( x - 2 )$

Use function notation to write $g$ in terms of $f ( x ) = x ^ { 3 }$ . $g ( x ) = - \frac { 1 } { 5 } ( x + 4 ) ^ { 3 }$

Given the following function, $h ( x )$ , find two functions $f$ and $g$ such that $( f \circ g ) ( x ) = h ( x )$ . $h ( x ) = \sqrt [ 3 ] { x ^ { 2 } - 11 }$

Use function notation to write $g$ in terms of $f ( x ) = x ^ { 3 }$ . $g ( x ) = - \frac { 1 } { 2 } ( x + 9 ) ^ { 3 }$

Find all real values of $x$ such that $f ( x ) = 0$ . $f ( x ) = \frac { - 3 x - 2 } { 5 }$

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

Determine the domain of $g ( x ) = \frac { 1 } { x ^ { 2 } - 81 }$ .

Find the domain of the function. $f ( t ) = \sqrt { 64 - t ^ { 2 } }$

Find the value(s) of $x$ for which $f ( x ) = g ( x )$ . $f ( x ) = x ^ { 2 } - 11 x - 36$ $g ( x ) = - 7 x - 4$

Find the inverse function of $f$ . $f ( x ) = x ^ { 5 } - 1$

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

Compare the graph of the following function with the graph of $f ( x ) = | x |$ . $y = \left| \frac { 7 } { 9 } x \right|$

Find $f \circ g$ . $f ( x ) = x + 3 \quad g ( x ) = \frac { 4 } { x ^ { 2 } - 9 }$

Find the slope and $y$ -intercept of the equation of the line. $y = - 2 x - 9$

Exam 1: Functions and Their Graphs

Restrict the domain of the following function fff so that the function is one-to-one and has an inverse function. f(x) =−∣x−4∣+2f ( x ) = - | x - 4 | + 2f(x) =−∣x−4∣+2

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.

If fff is an even function, determine if ggg is even, odd, or neither. g(x) =−f(x−2) g ( x ) = - f ( x - 2 ) g(x) =−f(x−2)

Use function notation to write ggg in terms of f(x) =x3f ( x ) = x ^ { 3 }f(x) =x3 . g(x) =−15(x+4) 3g ( x ) = - \frac { 1 } { 5 } ( x + 4 ) ^ { 3 }g(x) =−51​(x+4) 3

Given the following function, h(x) h ( x ) h(x) , find two functions fff and ggg such that (f∘g) (x) =h(x) ( f \circ g ) ( x ) = h ( x ) (f∘g) (x) =h(x) . h(x) =x2−113h ( x ) = \sqrt [ 3 ] { x ^ { 2 } - 11 }h(x) =3x2−11​

Use function notation to write ggg in terms of f(x) =x3f ( x ) = x ^ { 3 }f(x) =x3 . g(x) =−12(x+9) 3g ( x ) = - \frac { 1 } { 2 } ( x + 9 ) ^ { 3 }g(x) =−21​(x+9) 3

Find all real values of xxx such that f(x) =0f ( x ) = 0f(x) =0 . f(x) =−3x−25f ( x ) = \frac { - 3 x - 2 } { 5 }f(x) =5−3x−2​

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

Determine the domain of g(x) =1x2−81g ( x ) = \frac { 1 } { x ^ { 2 } - 81 }g(x) =x2−811​ .

Find the domain of the function. f(t) =64−t2f ( t ) = \sqrt { 64 - t ^ { 2 } }f(t) =64−t2​

Find the value(s) of xxx for which f(x) =g(x) f ( x ) = g ( x ) f(x) =g(x) . f(x) =x2−11x−36f ( x ) = x ^ { 2 } - 11 x - 36f(x) =x2−11x−36 g(x) =−7x−4g ( x ) = - 7 x - 4g(x) =−7x−4

Find the inverse function of fff . f(x) =x5−1f ( x ) = x ^ { 5 } - 1f(x) =x5−1

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

Graph the function and determine the interval(s) for which f(x) ≥0f ( x ) \geq 0f(x) ≥0 . f(x) =−x2+4xf ( x ) = - x ^ { 2 } + 4 xf(x) =−x2+4x

Use the graph of the function to find the domain and range of fff .

Compare the graph of the following function with the graph of f(x) =∣x∣f ( x ) = | x |f(x) =∣x∣ . y=∣79x∣y = \left| \frac { 7 } { 9 } x \right|y=​97​x​

Find f∘gf \circ gf∘g . f(x) =x+3g(x) =4x2−9f ( x ) = x + 3 \quad g ( x ) = \frac { 4 } { x ^ { 2 } - 9 }f(x) =x+3g(x) =x2−94​

Find the slope and yyy -intercept of the equation of the line. y=−2x−9y = - 2 x - 9y=−2x−9

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

Restrict the domain of the following function $f$ so that the function is one-to-one and has an inverse function. $f ( x ) = - | x - 4 | + 2$

If $f$ is an even function, determine if $g$ is even, odd, or neither. $g ( x ) = - f ( x - 2 )$

Use function notation to write $g$ in terms of $f ( x ) = x ^ { 3 }$ . $g ( x ) = - \frac { 1 } { 5 } ( x + 4 ) ^ { 3 }$

Given the following function, $h ( x )$ , find two functions $f$ and $g$ such that $( f \circ g ) ( x ) = h ( x )$ . $h ( x ) = \sqrt [ 3 ] { x ^ { 2 } - 11 }$

Use function notation to write $g$ in terms of $f ( x ) = x ^ { 3 }$ . $g ( x ) = - \frac { 1 } { 2 } ( x + 9 ) ^ { 3 }$

Find all real values of $x$ such that $f ( x ) = 0$ . $f ( x ) = \frac { - 3 x - 2 } { 5 }$

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

Determine the domain of $g ( x ) = \frac { 1 } { x ^ { 2 } - 81 }$ .

Find the domain of the function. $f ( t ) = \sqrt { 64 - t ^ { 2 } }$

Find the value(s) of $x$ for which $f ( x ) = g ( x )$ . $f ( x ) = x ^ { 2 } - 11 x - 36$ $g ( x ) = - 7 x - 4$

Find the inverse function of $f$ . $f ( x ) = x ^ { 5 } - 1$

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

Compare the graph of the following function with the graph of $f ( x ) = | x |$ . $y = \left| \frac { 7 } { 9 } x \right|$

Find $f \circ g$ . $f ( x ) = x + 3 \quad g ( x ) = \frac { 4 } { x ^ { 2 } - 9 }$

Find the slope and $y$ -intercept of the equation of the line. $y = - 2 x - 9$