Question 1

Estimate the slope of the line.

Accepted Answer

A)  $- 5$ 
B) 0 
C) 5 
D) $\frac { 1 } { 5 }$ 
E) $\frac { 2 } { 5 }$ 
A)  $- 5$ 
B) 0 
C) 5 
D) $\frac { 1 } { 5 }$ 
E) $\frac { 2 } { 5 }$ C

Question 2

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 C

Question 3

Compare the graph of the following function with the graph of $f ( x ) = \sqrt { x }$.
$$y = \sqrt { - x + 4 }$$

Accepted Answer

A)  First a vertical shift of 4 units up then a reflection in the $y$-axis. 
B)  First a horizontal shift of 4 units to the left then a reflection in the $y$-axis. 
C)  First a vertical shift of 4 units up then a reflection in the $x$-axis. 
D)  First a horizontal shift of 4 units to the left, then a vertical shift of 4 units up and then a reflection in the $y$-axis. 
E)  First a horizontal shift of 4 units to the left then a reflection in the $x$-axis. 
A)  First a vertical shift of 4 units up then a reflection in the $y$-axis. 
B)  First a horizontal shift of 4 units to the left then a reflection in the $y$-axis. 
C)  First a vertical shift of 4 units up then a reflection in the $x$-axis. 
D)  First a horizontal shift of 4 units to the left, then a vertical shift of 4 units up and then a reflection in the $y$-axis. 
E)  First a horizontal shift of 4 units to the left then a reflection in the $x$-axis. B

Question 4

Find $( f - g ) ( x )$
$$f ( x ) = - \frac { 8 x } { 4 x + 7 } \quad g ( x ) = - \frac { 4 } { x }$$

Accepted Answer

A)  $( f - g ) ( x ) = \frac { - 8 x + 4 } { 3 x + 7 }$ 
B)  $( f - g ) ( x ) = \frac { - 8 x + 23 } { 4 x + 7 }$ 
C)  $( f - g ) ( x ) = \frac { - 8 x + 9 } { 4 x + 7 }$ 
D) 
$( f - g ) ( x ) = \frac { - 8 x ^ { 2 } + 16 x - 28 } { 4 x ^ { 2 } + 7 x }$ 
E) 
$( f - g ) ( x ) = \frac { - 8 x ^ { 2 } + 16 x + 28 } { 4 x ^ { 2 } + 7 x }$ 
A)  $( f - g ) ( x ) = \frac { - 8 x + 4 } { 3 x + 7 }$ 
B)  $( f - g ) ( x ) = \frac { - 8 x + 23 } { 4 x + 7 }$ 
C)  $( f - g ) ( x ) = \frac { - 8 x + 9 } { 4 x + 7 }$ 
D) 
$( f - g ) ( x ) = \frac { - 8 x ^ { 2 } + 16 x - 28 } { 4 x ^ { 2 } + 7 x }$ 
E) 
$( f - g ) ( x ) = \frac { - 8 x ^ { 2 } + 16 x + 28 } { 4 x ^ { 2 } + 7 x }$

Question 5

Given that $f ( x ) = \sqrt [ 4 ] { x - 4 }$ and $g ( x ) = x ^ { 4 } + 4$ determine the value of the following (if possible).
$$( f \circ g ) ( 0 )$$

Accepted Answer

A)  0 
B)  2 
C)  4 
D)  $x ^ { 4 } - 16$ 
E)  not possible 
A)  0 
B)  2 
C)  4 
D)  $x ^ { 4 } - 16$ 
E)  not possible

Question 6

Evaluate the following function for $f ( x ) = - 2 x ^ { 2 } + 1$ and $g ( x ) = x + 4$ algebraically.
$$\left( \frac { f } { g } \right) ( q - 4 )$$

Accepted Answer

A)  $\frac { - 2 q ^ { 2 } + 5 } { q + 8 }$ 
B)  $\frac { - 2 q ^ { 2 } + 8 q - 31 } { q }$ 
C)  $\frac { - 2 q ^ { 2 } + 5 } { q }$ 
D)  $\frac { - 2 q ^ { 2 } + 16 q - 31 } { q }$ 
E)  $\frac { - 2 q ^ { 2 } - 3 } { q }$ 
A)  $\frac { - 2 q ^ { 2 } + 5 } { q + 8 }$ 
B)  $\frac { - 2 q ^ { 2 } + 8 q - 31 } { q }$ 
C)  $\frac { - 2 q ^ { 2 } + 5 } { q }$ 
D)  $\frac { - 2 q ^ { 2 } + 16 q - 31 } { q }$ 
E)  $\frac { - 2 q ^ { 2 } - 3 } { q }$

Question 7

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 8

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

Accepted Answer

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

Question 9

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

Question 10

Evaluate the function at the specified value of the independent variable and simplify. $$f ( p ) = \frac { - 3 p } { 4 p - 3 }$$
$f ( s + 8 )$

Accepted Answer

A)  $\frac { - 3 s - 24 } { 4 s + 29 }$ 
B)  $\frac { - 3 s + 24 } { 4 s + 29 }$ 
C)  $\frac { - 3 p - 24 } { 4 p + 29 }$ 
D)  $\frac { 24 } { 35 }$ 
E)  $- \frac { 24 } { 29 }$ 
A)  $\frac { - 3 s - 24 } { 4 s + 29 }$ 
B)  $\frac { - 3 s + 24 } { 4 s + 29 }$ 
C)  $\frac { - 3 p - 24 } { 4 p + 29 }$ 
D)  $\frac { 24 } { 35 }$ 
E)  $- \frac { 24 } { 29 }$

Question 11

Evaluate the function at the specified value of the independent variable and simplify. $$q ( p ) = \frac { - 2 p } { 5 p - 2 }$$
$q ( x - 9 )$

Accepted Answer

A)  $\frac { - 2 x + 18 } { 5 x - 47 }$ 
B)  $\frac { - 2 x - 18 } { 5 x - 47 }$ 
C)  $\frac { - 2 p + 18 } { 5 p - 47 }$ 
D)  $\frac { 18 } { 43 }$ 
E)  $- \frac { 18 } { 47 }$ 
A)  $\frac { - 2 x + 18 } { 5 x - 47 }$ 
B)  $\frac { - 2 x - 18 } { 5 x - 47 }$ 
C)  $\frac { - 2 p + 18 } { 5 p - 47 }$ 
D)  $\frac { 18 } { 43 }$ 
E)  $- \frac { 18 } { 47 }$

Question 12

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

Accepted Answer

A) 
$\begin{array}{rlrl}
f(g(x)) & =-\frac{5}{7}\left(\frac{7 x+21}{5}\right)-3 & g(f(x)) & =-\frac{7\left(-\frac{5}{7} x-3\right)+21}{5} \
& =\left(\frac{7 x+21}{7}\right)-3 & & =-\frac{(-5 x-21)+21}{5} \
& =(x+3)-3 & & =\frac{-5 x-21+21}{5} \
& =x+3-3 & &  \
& =x & &= \frac{5 x}{5}
\end{array}$ 
B) 
$\begin{array}{rlrl}
f(g(x)) & =-\frac{5}{7}\left(-\frac{7 x+21}{5}\right)-21 & g(f(x)) & =-\frac{7\left(-\frac{5}{7} x-3\right)+21}{5} \
& =\left(\frac{35 x+21}{35}\right)-21 & & =-\frac{(-5 x-3)+21}{5} \
& =(x+21)-21 & & =\frac{5 x+3-21}{5} \
& =x+21-21 & & =\frac{5 x}{5} \
& =x & &=x
\end{array}$ 
C) 
$\begin{array}{l}
\begin{array}{c}
f(g(x)) &=-\frac{5}{7}\left(-\frac{7 x+3}{5}\right)-3 \
&=\left(\frac{35 x+3}{35}\right)-3 \
&=(x+3)-3 \
&=x+3-3 \
&=x
\end{array}
\begin{array}{c}
g(f(x))=-\frac{7\left(-\frac{5}{7} x-3\right)+21}{5}\
=-\frac{(-5 x-3)+3}{5}\
=\frac{5 x+3-3}{5}\
=\frac{5 x}{5}\
=x
\end{array}
\end{array}$ 
D) 
$\begin{array}{l}
\begin{array}{lll}
f(g(x)) &=-\frac{5}{7}\left(-\frac{7 x+21}{5}\right)-3 \
&=\left(\frac{7 x+21}{7}\right)-3 \
&=(x+3)-3 \
&=x+3-3 \
&=x
\end{array}
\begin{array}{lll}
g(f(x)) &=-\frac{7\left(-\frac{5}{7} x-3\right)+21}{5} \
&=-\frac{(-5 x-21)+21}{5} \
&=\frac{5 x+21-21}{5} \
&=\quad \frac{5 x}{5} \
&=x
\end{array}
\end{array}$ 
E) 
$\begin{array}{l}
\begin{array}{lll}
f(g(x)) &=-\frac{7}{5}\left(-\frac{5 x+15}{7}\right)-3 \
&=\left(\frac{5 x+15}{5}\right)-3 \
&=(x+3)-3 \
&=x+3-3 \
&=x
\end{array}
\begin{array}{lll}
g(f(x)) &=-\frac{7\left(-\frac{5}{7} x-3\right)+21}{5} \
&=-\frac{(-5 x-3)+21}{35} \
&=\frac{5 x+3-21}{35} \
&=\quad \frac{35 x}{35} \
&=\quad x
\end{array}
\end{array}$ 
A) 
$\begin{array}{rlrl}
f(g(x)) & =-\frac{5}{7}\left(\frac{7 x+21}{5}\right)-3 & g(f(x)) & =-\frac{7\left(-\frac{5}{7} x-3\right)+21}{5} \
& =\left(\frac{7 x+21}{7}\right)-3 & & =-\frac{(-5 x-21)+21}{5} \
& =(x+3)-3 & & =\frac{-5 x-21+21}{5} \
& =x+3-3 & &  \
& =x & &= \frac{5 x}{5}
\end{array}$ 
B) 
$\begin{array}{rlrl}
f(g(x)) & =-\frac{5}{7}\left(-\frac{7 x+21}{5}\right)-21 & g(f(x)) & =-\frac{7\left(-\frac{5}{7} x-3\right)+21}{5} \
& =\left(\frac{35 x+21}{35}\right)-21 & & =-\frac{(-5 x-3)+21}{5} \
& =(x+21)-21 & & =\frac{5 x+3-21}{5} \
& =x+21-21 & & =\frac{5 x}{5} \
& =x & &=x
\end{array}$ 
C) 
$\begin{array}{l}
\begin{array}{c}
f(g(x)) &=-\frac{5}{7}\left(-\frac{7 x+3}{5}\right)-3 \
&=\left(\frac{35 x+3}{35}\right)-3 \
&=(x+3)-3 \
&=x+3-3 \
&=x
\end{array}
\begin{array}{c}
g(f(x))=-\frac{7\left(-\frac{5}{7} x-3\right)+21}{5}\
=-\frac{(-5 x-3)+3}{5}\
=\frac{5 x+3-3}{5}\
=\frac{5 x}{5}\
=x
\end{array}
\end{array}$ 
D) 
$\begin{array}{l}
\begin{array}{lll}
f(g(x)) &=-\frac{5}{7}\left(-\frac{7 x+21}{5}\right)-3 \
&=\left(\frac{7 x+21}{7}\right)-3 \
&=(x+3)-3 \
&=x+3-3 \
&=x
\end{array}
\begin{array}{lll}
g(f(x)) &=-\frac{7\left(-\frac{5}{7} x-3\right)+21}{5} \
&=-\frac{(-5 x-21)+21}{5} \
&=\frac{5 x+21-21}{5} \
&=\quad \frac{5 x}{5} \
&=x
\end{array}
\end{array}$ 
E) 
$\begin{array}{l}
\begin{array}{lll}
f(g(x)) &=-\frac{7}{5}\left(-\frac{5 x+15}{7}\right)-3 \
&=\left(\frac{5 x+15}{5}\right)-3 \
&=(x+3)-3 \
&=x+3-3 \
&=x
\end{array}
\begin{array}{lll}
g(f(x)) &=-\frac{7\left(-\frac{5}{7} x-3\right)+21}{5} \
&=-\frac{(-5 x-3)+21}{35} \
&=\frac{5 x+3-21}{35} \
&=\quad \frac{35 x}{35} \
&=\quad x
\end{array}
\end{array}$

Question 13

Determine whether the function is even, odd, or neither. $$f ( x ) = 4 x ^ { 3 } - 2 x$$

Accepted Answer

A)  neither 
B)  even 
C)  odd 
A)  neither 
B)  even 
C)  odd

Question 14

Compare the graph of the following function with the graph of $f ( x ) = x ^ { 3 }$.
$$y = [ 5 ( x + 10 ) ] ^ { 3 }$$

Accepted Answer

A)  vertical shift of 10 units up 
B)  vertical shift of 10 units up horizontal shrink of $\frac { 1 } { 5 }$ units 
C)  horizontal shift of 10 units to the left horizontal shrink of $\frac { 1 } { 125 }$ units 
D)  horizontal shift of 10 units to the left horizontal stretch of $\frac { 1 } { 5 }$ units 
E)  horizontal shift of 10 units to the left vertical shift of 5 units up 
A)  vertical shift of 10 units up 
B)  vertical shift of 10 units up horizontal shrink of $\frac { 1 } { 5 }$ units 
C)  horizontal shift of 10 units to the left horizontal shrink of $\frac { 1 } { 125 }$ units 
D)  horizontal shift of 10 units to the left horizontal stretch of $\frac { 1 } { 5 }$ units 
E)  horizontal shift of 10 units to the left vertical shift of 5 units up

Question 15

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)  $f ( x ) = - | x - 1 | + 1$ 
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)  $f ( x ) = - | x - 1 | + 1$

Question 16

Evaluate the indicated function for $f ( x ) = x ^ { 2 } - 1$ and $g ( x ) = x - 6$.
$( f g ) ( - 2 )$

Accepted Answer

A)  $- 24$ 
B)  40 
C)  $- 2$ 
D)  12 
E)  24 
A)  $- 24$ 
B)  40 
C)  $- 2$ 
D)  12 
E)  24

Question 17

Estimate the slope of the line.

Accepted Answer

A)  $- 5$ 
B) 0 
C) 5 
D) $\frac { 1 } { 5 }$ 
E) $\frac { 2 } { 5 }$ 
A)  $- 5$ 
B) 0 
C) 5 
D) $\frac { 1 } { 5 }$ 
E) $\frac { 2 } { 5 }$

Question 18

Find the difference quotient and simplify your answer. $$f ( w ) = - 9 w ^ { 2 } + 2 w , \quad \frac { f ( 4 + h ) - f ( 4 ) } { h } , h \neq 0$$

Accepted Answer

A)  $10 + h$ 
B)  $- 70 - 9 w + \frac { 16 } { w }$ 
C)  $2 - 9 w + \frac { 16 } { w }$ 
D)  $2 - 9 h$ 
E)  $- 70 - 9 h$ 
A)  $10 + h$ 
B)  $- 70 - 9 w + \frac { 16 } { w }$ 
C)  $2 - 9 w + \frac { 16 } { w }$ 
D)  $2 - 9 h$ 
E)  $- 70 - 9 h$

Question 19

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 20

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)

Estimate the slope of the line.

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

Compare the graph of the following function with the graph of $f ( x ) = \sqrt { x }$ . $y = \sqrt { - x + 4 }$

Find $( f - g ) ( x )$ $f ( x ) = - \frac { 8 x } { 4 x + 7 } \quad g ( x ) = - \frac { 4 } { x }$

Given that $f ( x ) = \sqrt [ 4 ] { x - 4 }$ and $g ( x ) = x ^ { 4 } + 4$ determine the value of the following (if possible). $( f \circ g ) ( 0 )$

Evaluate the following function for $f ( x ) = - 2 x ^ { 2 } + 1$ and $g ( x ) = x + 4$ algebraically. $\left( \frac { f } { g } \right) ( q - 4 )$

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

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

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

Evaluate the function at the specified value of the independent variable and simplify. $f ( p ) = \frac { - 3 p } { 4 p - 3 }$ $f ( s + 8 )$

Evaluate the function at the specified value of the independent variable and simplify. $q ( p ) = \frac { - 2 p } { 5 p - 2 }$ $q ( x - 9 )$

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

Determine whether the function is even, odd, or neither. $f ( x ) = 4 x ^ { 3 } - 2 x$

Compare the graph of the following function with the graph of $f ( x ) = x ^ { 3 }$ . $y = [ 5 ( x + 10 ) ] ^ { 3 }$

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

Evaluate the indicated function for $f ( x ) = x ^ { 2 } - 1$ and $g ( x ) = x - 6$ . $( f g ) ( - 2 )$

Estimate the slope of the line.

Find the difference quotient and simplify your answer. $f ( w ) = - 9 w ^ { 2 } + 2 w , \quad \frac { f ( 4 + h ) - f ( 4 ) } { h } , h \neq 0$

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

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

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

Calculus Practice Problems

Filters

Exam 1: Functions and Their Graphs

Estimate the slope of the line.

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)

Compare the graph of the following function with the graph of f(x)=xf ( x ) = \sqrt { x }f(x)=x​ . y=−x+4y = \sqrt { - x + 4 }y=−x+4​

Find (f−g)(x)( f - g ) ( x )(f−g)(x) f(x)=−8x4x+7g(x)=−4xf ( x ) = - \frac { 8 x } { 4 x + 7 } \quad g ( x ) = - \frac { 4 } { x }f(x)=−4x+78x​g(x)=−x4​

Given that f(x)=x−44f ( x ) = \sqrt [ 4 ] { x - 4 }f(x)=4x−4​ and g(x)=x4+4g ( x ) = x ^ { 4 } + 4g(x)=x4+4 determine the value of the following (if possible). (f∘g)(0)( f \circ g ) ( 0 )(f∘g)(0)

Evaluate the following function for f(x)=−2x2+1f ( x ) = - 2 x ^ { 2 } + 1f(x)=−2x2+1 and g(x)=x+4g ( x ) = x + 4g(x)=x+4 algebraically. (fg)(q−4)\left( \frac { f } { g } \right) ( q - 4 )(gf​)(q−4)

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

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

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

Evaluate the function at the specified value of the independent variable and simplify. f(p)=−3p4p−3f ( p ) = \frac { - 3 p } { 4 p - 3 }f(p)=4p−3−3p​ f(s+8)f ( s + 8 )f(s+8)

Evaluate the function at the specified value of the independent variable and simplify. q(p)=−2p5p−2q ( p ) = \frac { - 2 p } { 5 p - 2 }q(p)=5p−2−2p​ q(x−9)q ( x - 9 )q(x−9)

Show algebraically that the functions fff and ggg shown below are inverse functions. 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​

Determine whether the function is even, odd, or neither. f(x)=4x3−2xf ( x ) = 4 x ^ { 3 } - 2 xf(x)=4x3−2x

Compare the graph of the following function with the graph of f(x)=x3f ( x ) = x ^ { 3 }f(x)=x3 . y=[5(x+10)]3y = [ 5 ( x + 10 ) ] ^ { 3 }y=[5(x+10)]3

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

Evaluate the indicated function for f(x)=x2−1f ( x ) = x ^ { 2 } - 1f(x)=x2−1 and g(x)=x−6g ( x ) = x - 6g(x)=x−6 . (fg)(−2)( f g ) ( - 2 )(fg)(−2)

Estimate the slope of the line.

Find the difference quotient and simplify your answer. f(w)=−9w2+2w,f(4+h)−f(4)h,h≠0f ( w ) = - 9 w ^ { 2 } + 2 w , \quad \frac { f ( 4 + h ) - f ( 4 ) } { h } , h \neq 0f(w)=−9w2+2w,hf(4+h)−f(4)​,h=0

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

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

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

Calculus Practice Problems

Filters

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

Compare the graph of the following function with the graph of $f ( x ) = \sqrt { x }$ . $y = \sqrt { - x + 4 }$

Find $( f - g ) ( x )$ $f ( x ) = - \frac { 8 x } { 4 x + 7 } \quad g ( x ) = - \frac { 4 } { x }$

Given that $f ( x ) = \sqrt [ 4 ] { x - 4 }$ and $g ( x ) = x ^ { 4 } + 4$ determine the value of the following (if possible). $( f \circ g ) ( 0 )$

Evaluate the following function for $f ( x ) = - 2 x ^ { 2 } + 1$ and $g ( x ) = x + 4$ algebraically. $\left( \frac { f } { g } \right) ( q - 4 )$

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

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

Evaluate the function at the specified value of the independent variable and simplify. $f ( p ) = \frac { - 3 p } { 4 p - 3 }$ $f ( s + 8 )$

Evaluate the function at the specified value of the independent variable and simplify. $q ( p ) = \frac { - 2 p } { 5 p - 2 }$ $q ( x - 9 )$

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

Determine whether the function is even, odd, or neither. $f ( x ) = 4 x ^ { 3 } - 2 x$

Compare the graph of the following function with the graph of $f ( x ) = x ^ { 3 }$ . $y = [ 5 ( x + 10 ) ] ^ { 3 }$

Evaluate the indicated function for $f ( x ) = x ^ { 2 } - 1$ and $g ( x ) = x - 6$ . $( f g ) ( - 2 )$

Find the difference quotient and simplify your answer. $f ( w ) = - 9 w ^ { 2 } + 2 w , \quad \frac { f ( 4 + h ) - f ( 4 ) } { h } , h \neq 0$

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