Question 1

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

Question 2

Show algebraically that the functions $ f $ and $ g $ shown below are inverse
$f(x)=-\frac{5}{7} x-3, \quad g(x)=-\frac{7 x+21}{5}$

Accepted Answer

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

Question 3

If $f$ is an even function, determine if $g$ is even, odd, or neither. $g ( x ) = - f ( x + 3 )$

Accepted Answer

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

Question 4

Find $f \circ g$.
$f ( x ) = \left| x ^ { 2 } - 6 \right| \quad g ( x ) = - 9 - x$

Accepted Answer

A)  $( f \circ g ) ( x ) = \left| x ^ { 2 } + 18 x + 75 \right|$ 
B)  $( f \circ g ) ( x ) = \left| x ^ { 2 } + 75 \right|$ 
C)  $( f \circ g ) ( x ) = \left| - 3 - x ^ { 2 } \right|$ 
D)  $( f \circ g ) ( x ) = \left| - 15 - x ^ { 2 } \right|$ 
E)  $( f \circ g ) ( x ) = - 9 - \left| x ^ { 2 } - 6 \right|$ 
A)  $( f \circ g ) ( x ) = \left| x ^ { 2 } + 18 x + 75 \right|$ 
B)  $( f \circ g ) ( x ) = \left| x ^ { 2 } + 75 \right|$ 
C)  $( f \circ g ) ( x ) = \left| - 3 - x ^ { 2 } \right|$ 
D)  $( f \circ g ) ( x ) = \left| - 15 - x ^ { 2 } \right|$ 
E)  $( f \circ g ) ( x ) = - 9 - \left| x ^ { 2 } - 6 \right|$

Question 5

Determine algebraically whether the following function is one-to-one $| x - 5 | , \quad x \leq 5$

Accepted Answer

A) 
$\begin{aligned}
| a - 5 | & = | b - 5 | \
5 - a & = 5 - b \quad ; \text { one-to-one } \
- a & = - b \quad \
a & = b
\end{aligned}$ 
B) 
$\begin{aligned}
a & = b \
| a - 5 | & = | b - 5 | \
| a | - 5 & = | b | - 5 \
| a | & = | b | \quad \text { one-to-one } \
a & = b
\end{aligned}$ 
C) 
$\begin{aligned}
| a - 5 | & = | b - 5 | \
a + 5 & = 5 - b \quad ; \text { not one-to-one } \
a & = - b
\end{aligned}$ 
D) 
$\begin{aligned}
| a - 5 | & = | b - 5 | \
| - 5 | - a & = | - 5 | + b ; \text { not one-to-one }
\end{aligned}$ 
E) 
$\begin{aligned}
- a & = b \
| a - 5 | & = | b - 5 | \
| - 5 | - a & = | - 5 | - b ; \text { one-to-one } \
- a & = - b \
a & = b
\end{aligned}$ 
A) 
$\begin{aligned}
| a - 5 | & = | b - 5 | \
5 - a & = 5 - b \quad ; \text { one-to-one } \
- a & = - b \quad \
a & = b
\end{aligned}$ 
B) 
$\begin{aligned}
a & = b \
| a - 5 | & = | b - 5 | \
| a | - 5 & = | b | - 5 \
| a | & = | b | \quad \text { one-to-one } \
a & = b
\end{aligned}$ 
C) 
$\begin{aligned}
| a - 5 | & = | b - 5 | \
a + 5 & = 5 - b \quad ; \text { not one-to-one } \
a & = - b
\end{aligned}$ 
D) 
$\begin{aligned}
| a - 5 | & = | b - 5 | \
| - 5 | - a & = | - 5 | + b ; \text { not one-to-one }
\end{aligned}$ 
E) 
$\begin{aligned}
- a & = b \
| a - 5 | & = | b - 5 | \
| - 5 | - a & = | - 5 | - b ; \text { one-to-one } \
- a & = - b \
a & = b
\end{aligned}$

Question 6

Determine the domain and range of the inverse function $f ^ { - 1 }$ of the following function $f$.
$f ( x ) = - | x + 6 | + 2$, where $x > - 6$

Accepted Answer

A)  Domain: $[ - 6 , \infty )$; Range: $[ 2 , \infty )$ 
B)  Domain: $( - \infty , 2 ]$; Range: $[ - 6 , \infty )$ 
C)  Domain: $[ - 6,2 ]$; Range: $[ - 6 , \infty )$ 
D)  Domain: $( - \infty , - 6 ]$; Range: $[ - 2 , \infty )$ 
E)  Domain: $( - \infty , \infty )$; Range: $( - \infty , \infty )$ 
A)  Domain: $[ - 6 , \infty )$; Range: $[ 2 , \infty )$ 
B)  Domain: $( - \infty , 2 ]$; Range: $[ - 6 , \infty )$ 
C)  Domain: $[ - 6,2 ]$; Range: $[ - 6 , \infty )$ 
D)  Domain: $( - \infty , - 6 ]$; Range: $[ - 2 , \infty )$ 
E)  Domain: $( - \infty , \infty )$; Range: $( - \infty , \infty )$

Question 7

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 8

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

Accepted Answer

A)  domain : all real numbers
range: $( - \infty , - 2 ) \cup ( - 1 , \infty )$ 
B)  domain : all real numbers
range : all real numbers 
C)  domain: $( - \infty , - 2 ) \cup ( - 2 , \infty )$
range: $( - \infty , - 2 ) \cup ( - 1 , \infty )$ 
D)  domain: $( - \infty , - 2 ) \cup ( - 1 , \infty )$
range : $( - \infty , - 2 ) \cup ( - 2 , \infty )$ 
E)  Domain: all real numbers
Range: $( - \infty , - 2 ] \cup [ - 1 , \infty )$ 
A)  domain : all real numbers
range: $( - \infty , - 2 ) \cup ( - 1 , \infty )$ 
B)  domain : all real numbers
range : all real numbers 
C)  domain: $( - \infty , - 2 ) \cup ( - 2 , \infty )$
range: $( - \infty , - 2 ) \cup ( - 1 , \infty )$ 
D)  domain: $( - \infty , - 2 ) \cup ( - 1 , \infty )$
range : $( - \infty , - 2 ) \cup ( - 2 , \infty )$ 
E)  Domain: all real numbers
Range: $( - \infty , - 2 ] \cup [ - 1 , \infty )$

Question 9

Determine whether lines $L _ { 1 }$ and $L _ { 2 }$ passing through the pairs of points are parallel, perpendicular, or neither.
$L _ { 1 } : ( 7 , - 4 ) , ( - 9 , - 1 )$
$L _ { 2 } : ( 4 , - 6 ) , ( - 3,9 )$

Accepted Answer

A)  parallel 
B)  perpendicular 
C)  neither 
A)  parallel 
B)  perpendicular 
C)  neither

Question 10

Compare the graph of the following function with the graph of $f ( x ) = | x |$.
$y = \left| \frac { 1 } { 9 } x \right|$

Accepted Answer

A)  vertical shift of $\frac { 1 } { 9 }$ unit up 
B)  horizontal stretch of $\frac { 9 } { 1 }$ unit 
C)  vertical shrink of $\frac { 1 } { 9 }$ unit 
D)  horizontal shrink of $\frac { 1 } { 9 }$ unit vertical shift of $\frac { 9 } { 1 }$ unit 
E)  horizontal shrink of $\frac { 1 } { 9 }$ unit 
A)  vertical shift of $\frac { 1 } { 9 }$ unit up 
B)  horizontal stretch of $\frac { 9 } { 1 }$ unit 
C)  vertical shrink of $\frac { 1 } { 9 }$ unit 
D)  horizontal shrink of $\frac { 1 } { 9 }$ unit vertical shift of $\frac { 9 } { 1 }$ unit 
E)  horizontal shrink of $\frac { 1 } { 9 }$ unit

Question 11

Use the functions $f ( x ) = \frac { 1 } { 125 } x - 5$ and $g ( x ) = x ^ { 3 }$ to find $( f \circ g ) ^ { - 1 }$.

Accepted Answer

A)  $( f \circ g ) ^ { - 1 } = \frac { x ^ { 3 } + 5 } { 5 }$ 
B)  $( f \circ g ) ^ { - 1 } = \frac { x ^ { 3 } - 625 } { 125 }$ 
C)  $( f \circ g ) ^ { - 1 } = \frac { \sqrt [ 3 ] { x + 5 } } { 5 }$ 
D)  $( f \circ g ) ^ { - 1 } = 5 x + 5$ 
E)  $( f \circ g ) ^ { - 1 } = 5 \sqrt [ 3 ] { x + 5 }$ 
A)  $( f \circ g ) ^ { - 1 } = \frac { x ^ { 3 } + 5 } { 5 }$ 
B)  $( f \circ g ) ^ { - 1 } = \frac { x ^ { 3 } - 625 } { 125 }$ 
C)  $( f \circ g ) ^ { - 1 } = \frac { \sqrt [ 3 ] { x + 5 } } { 5 }$ 
D)  $( f \circ g ) ^ { - 1 } = 5 x + 5$ 
E)  $( f \circ g ) ^ { - 1 } = 5 \sqrt [ 3 ] { x + 5 }$

Question 12

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)  $f ( x ) = - 1 - \sqrt { 1 + x }$ 
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)  $f ( x ) = - 1 - \sqrt { 1 + x }$

Question 13

Find the domain of the function.
$g ( p ) = \sqrt { 4 - p ^ { 2 } }$

Accepted Answer

A)  $- 2 \leq p \leq 2$ 
B)  $p \leq - 2$ or $p \geq 2$ 
C)  $p \geq 0$ 
D)  $p \leq 2$ 
E)  all real numbers 
A)  $- 2 \leq p \leq 2$ 
B)  $p \leq - 2$ or $p \geq 2$ 
C)  $p \geq 0$ 
D)  $p \leq 2$ 
E)  all real numbers

Question 14

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.
$\begin{array}{|c|c|}
\hline \text { Force, } \boldsymbol{F} & \text { Elongation, } \boldsymbol{d} \
\hline 20 & 3.5 \
\hline 40 & 6.3 \
\hline 60 & 10.0 \
\hline 80 & 13.3 \
\hline 100 & 16.5 \
\hline
\end{array}$

Find the equation of the line that seems to best fit the data.

Accepted Answer

A)  $\quad F = 12.098 d$ 
B)  $F = 3.024 d$ 
C)  $\quad F = 6.049 d$ 
D)  $\quad F = 4.537 d$ 
E)  $\quad F = 7.561 d$ 
A)  $\quad F = 12.098 d$ 
B)  $F = 3.024 d$ 
C)  $\quad F = 6.049 d$ 
D)  $\quad F = 4.537 d$ 
E)  $\quad F = 7.561 d$

Question 15

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)  $f ( x ) = - 1 + \sqrt { 1 + x }$ 
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)  $f ( x ) = - 1 + \sqrt { 1 + x }$

Question 16

Determine whether lines $L _ { 1 }$ and $L _ { 2 }$ passing through the pairs of points are parallel, perpendicular, or neither.
$L _ { 1 } : ( - 1,1 ) , ( - 1 , - 6 )$
$L _ { 2 } : ( 3 , - 8 ) , ( 24 , - 8 )$

Accepted Answer

A)  parallel 
B)  perpendicular 
C)  neither 
A)  parallel 
B)  perpendicular 
C)  neither

Question 17

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

Accepted Answer

A)  $x = 6 y - 9$ 
B)  $x = - 5$ 
C)  $y = x + 5$ 
D)  $y = \left| - 1 - x ^ { 2 } \right|$ 
E)  $y = \sqrt { - 5 + 4 x }$ 
A)  $x = 6 y - 9$ 
B)  $x = - 5$ 
C)  $y = x + 5$ 
D)  $y = \left| - 1 - x ^ { 2 } \right|$ 
E)  $y = \sqrt { - 5 + 4 x }$

Question 18

Algebraically determine whether the function below is even, odd, or neither.
$f ( q ) = 2 q ^ { 3 / 2 }$

Accepted Answer

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

Question 19

Show algebraically that the functions $ f $ and $ g $ shown below are inverse functions.

Accepted Answer

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

Question 20

Find all real values of $x$ such that $f ( x ) = 0$.
$f ( x ) = \frac { 7 x - 5 } { 7 }$

Accepted Answer

A)  $\frac { 5 } { 49 }$ 
B)  $\pm \frac { 5 } { 49 }$ 
C)  $\pm \frac { 5 } { 7 }$ 
D)  $\frac { 5 } { 7 }$ 
E)  $- \frac { 5 } { 7 }$ 
A)  $\frac { 5 } { 49 }$ 
B)  $\pm \frac { 5 } { 49 }$ 
C)  $\pm \frac { 5 } { 7 }$ 
D)  $\frac { 5 } { 7 }$ 
E)  $- \frac { 5 } { 7 }$

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

Show algebraically that the functions $f$ and $g$ shown below are inverse $f(x)=-\frac{5}{7} x-3, \quad g(x)=-\frac{7 x+21}{5}$

If $f$ is an even function, determine if $g$ is even, odd, or neither. $g ( x ) = - f ( x + 3 )$

Find $f \circ g$ . $f ( x ) = \left| x ^ { 2 } - 6 \right| \quad g ( x ) = - 9 - x$

Determine algebraically whether the following function is one-to-one $| x - 5 | , \quad x \leq 5$

Determine the domain and range of the inverse function $f ^ { - 1 }$ of the following function $f$ . $f ( x ) = - | x + 6 | + 2$ , where $x > - 6$

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

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

Determine whether lines $L _ { 1 }$ and $L _ { 2 }$ passing through the pairs of points are parallel, perpendicular, or neither. $L _ { 1 } : ( 7 , - 4 ) , ( - 9 , - 1 )$ $L _ { 2 } : ( 4 , - 6 ) , ( - 3,9 )$

Compare the graph of the following function with the graph of $f ( x ) = | x |$ . $y = \left| \frac { 1 } { 9 } x \right|$

Use the functions $f ( x ) = \frac { 1 } { 125 } x - 5$ and $g ( x ) = x ^ { 3 }$ to find $( f \circ g ) ^ { - 1 }$ .

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

Find the domain of the function. $g ( p ) = \sqrt { 4 - p ^ { 2 } }$

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

Determine whether lines $L _ { 1 }$ and $L _ { 2 }$ passing through the pairs of points are parallel, perpendicular, or neither. $L _ { 1 } : ( - 1,1 ) , ( - 1 , - 6 )$ $L _ { 2 } : ( 3 , - 8 ) , ( 24 , - 8 )$

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

Algebraically determine whether the function below is even, odd, or neither. $f ( q ) = 2 q ^ { 3 / 2 }$

Show algebraically that the functions $f$ and $g$ shown below are inverse functions.

Find all real values of $x$ such that $f ( x ) = 0$ . $f ( x ) = \frac { 7 x - 5 } { 7 }$

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

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Filters

Exam 1: Functions and Their Graphs

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

Show algebraically that the functions f f f and g g g shown below are inverse f(x)=−57x−3,g(x)=−7x+215f(x)=-\frac{5}{7} x-3, \quad g(x)=-\frac{7 x+21}{5}f(x)=−75​x−3,g(x)=−57x+21​

If fff is an even function, determine if ggg is even, odd, or neither. g(x)=−f(x+3)g ( x ) = - f ( x + 3 )g(x)=−f(x+3)

Find f∘gf \circ gf∘g . f(x)=∣x2−6∣g(x)=−9−xf ( x ) = \left| x ^ { 2 } - 6 \right| \quad g ( x ) = - 9 - xf(x)=​x2−6​g(x)=−9−x

Determine algebraically whether the following function is one-to-one ∣x−5∣,x≤5| x - 5 | , \quad x \leq 5∣x−5∣,x≤5

Determine the domain and range of the inverse function f−1f ^ { - 1 }f−1 of the following function fff . f(x)=−∣x+6∣+2f ( x ) = - | x + 6 | + 2f(x)=−∣x+6∣+2 , where x>−6x > - 6x>−6

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

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=∣19x∣y = \left| \frac { 1 } { 9 } x \right|y=​91​x​

Use the functions f(x)=1125x−5f ( x ) = \frac { 1 } { 125 } x - 5f(x)=1251​x−5 and g(x)=x3g ( x ) = x ^ { 3 }g(x)=x3 to find (f∘g)−1( f \circ g ) ^ { - 1 }(f∘g)−1 .

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

Find the domain of the function. g(p)=4−p2g ( p ) = \sqrt { 4 - p ^ { 2 } }g(p)=4−p2​

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

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

Algebraically determine whether the function below is even, odd, or neither. f(q)=2q3/2f ( q ) = 2 q ^ { 3 / 2 }f(q)=2q3/2

Show algebraically that the functions f f f and g g g shown below are inverse functions.

Find all real values of xxx such that f(x)=0f ( x ) = 0f(x)=0 . f(x)=7x−57f ( x ) = \frac { 7 x - 5 } { 7 }f(x)=77x−5​

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

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Filters

Show algebraically that the functions $f$ and $g$ shown below are inverse $f(x)=-\frac{5}{7} x-3, \quad g(x)=-\frac{7 x+21}{5}$

If $f$ is an even function, determine if $g$ is even, odd, or neither. $g ( x ) = - f ( x + 3 )$

Find $f \circ g$ . $f ( x ) = \left| x ^ { 2 } - 6 \right| \quad g ( x ) = - 9 - x$

Determine algebraically whether the following function is one-to-one $| x - 5 | , \quad x \leq 5$

Determine the domain and range of the inverse function $f ^ { - 1 }$ of the following function $f$ . $f ( x ) = - | x + 6 | + 2$ , where $x > - 6$

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

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

Compare the graph of the following function with the graph of $f ( x ) = | x |$ . $y = \left| \frac { 1 } { 9 } x \right|$

Use the functions $f ( x ) = \frac { 1 } { 125 } x - 5$ and $g ( x ) = x ^ { 3 }$ to find $( f \circ g ) ^ { - 1 }$ .

Find the domain of the function. $g ( p ) = \sqrt { 4 - p ^ { 2 } }$

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

Algebraically determine whether the function below is even, odd, or neither. $f ( q ) = 2 q ^ { 3 / 2 }$

Show algebraically that the functions $f$ and $g$ shown below are inverse functions.

Find all real values of $x$ such that $f ( x ) = 0$ . $f ( x ) = \frac { 7 x - 5 } { 7 }$