Question 1

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 3$ and $g ( x ) = x ^ { 2 }$ to find the value $\left( f ^ { - 1 } \circ g ^ { - 1 } \right) ( 1 )$
A)32
B)35
C)33
D)37
E)30

Accepted Answer

To solve $\left( f ^ { - 1 } \circ g ^ { - 1 } \right) ( 1 )$, we first find $g^{-1}(x)$ and then $f^{-1}(x)$, and finally apply these inverses to 1 in the correct order.1. Find $g^{-1}(x)$: Given $g(x) = x^2$, to find $g^{-1}(x)$, we solve $y = x^2$ for $x$, giving $x = \sqrt{y}$. So, $g^{-1}(x) = \sqrt{x}$.2. Find $f^{-1}(x)$: Given $f(x) = \frac{1}{8}x - 3$, to find $f^{-1}(x)$, we solve $y = \frac{1}{8}x - 3$ for $x$, giving $x = 8(y + 3)$. So, $f^{-1}(x) = 8(x + 3)$.3. Apply $g^{-1}$ to 1: $g^{-1}(1) = \sqrt{1} = 1$.4. Apply $f^{-1}$ to the result: $f^{-1}(1) = 8(1 + 3) = 8 \times 4 = 32$.Therefore, $\left( f ^ { - 1 } \circ g ^ { - 1 } \right) ( 1 ) = 32$.

Question 2

Show that $f$ and $g$ are functions by using the definition of inverse functions. $f ( x ) = \frac { 1 } { x } , g ( x ) = \frac { 1 } { x }$
A) $f ( g ( x ) ) = x , g ( f ( x ) ) = x - 1$
B) $f ( g ( x ) ) = x - 1 , g ( f ( x ) ) = x$
C) $f ( g ( x ) ) = x , g ( f ( x ) ) = x$
D) $f ( g ( x ) ) = x , g ( f ( x ) ) = x ^ { - 1 }$
E) $f ( g ( x ) ) = x ^ { - 1 } , g ( f ( x ) ) = x$

Accepted Answer

For two functions to be inverses of each other, the composition of one with the other in both orders must result in the identity function, i.e., $f(g(x)) = x$ and $g(f(x)) = x$. Given $f(x) = \frac{1}{x}$ and $g(x) = \frac{1}{x}$, composing $f$ with $g$ (or $g$ with $f$) yields $f(g(x)) = f(\frac{1}{x}) = \frac{1}{\frac{1}{x}} = x$ and similarly, $g(f(x)) = g(\frac{1}{x}) = \frac{1}{\frac{1}{x}} = x$, confirming that $f$ and $g$ are indeed inverse functions of each other.

Question 3

Use the functions given by $f ( x ) = x + 4$ and $g ( x ) = 2 x - 6$ to find the composition of functions $g ^ { - 1 } \circ f ^ { - 1 }$
A) $\frac { x - 2 } { 2 }$
B) $\frac { x + 2 } { - 2 }$
C) $\frac { x - 2 } { - 2 }$
D) $\frac { - ( x + 2 ) } { 2 }$
E) $\frac { x + 2 } { 2 }$

Accepted Answer

E

Question 4

Determine whether the function has an inverse function, If it does, find its inverse function. $f ( x ) = | x - 2 | , x \leq 2$
A)No, $f$ does not have an inverse function.
B)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = 2 - x , x \geq 0$
C)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = - 2 - x , x \geq 0$
D)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = 2 + x , x \geq 0$
E)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = - 2 + x , x \geq 0$

Accepted Answer

Given $f(x) = |x - 2|$ for $x \leq 2$, the function simplifies to $f(x) = 2 - x$ because for $x \leq 2$, $x - 2 \leq 0$ and the absolute value of a negative number is its positive counterpart. The inverse function swaps $x$ and $y$, leading to $x = 2 - y$, which simplifies to $y = 2 - x$. Since $f(x)$ is decreasing and passes the horizontal line test, it has an inverse, which is $f^{-1}(x) = 2 - x$, valid for $x \geq 0$ because $f(x)$ outputs non-negative values.

Question 5

Use the functions given by $f ( x ) = x + 4$ and $g ( x ) = 2 x - 6$ to find the composition of functions $(g\circ f)^{-1}$
A) $\frac { - ( x + 2 ) } { - 2 }$
B) $\frac { - x + 2 } { 2 }$
C) $\frac { x - 2 } { 2 }$
D) $\frac { x - 2 } { - 2 }$
E) $\frac { x + 2 } { 2 }$

Accepted Answer

First, find the composition $g(f(x)) = 2(x+4)-6 = 2x+2$. Then, to find the inverse, swap $x$ and $y$ and solve for $y$, giving $x = 2y + 2$, which simplifies to $y = \frac{x - 2}{2}$.

Question 6

Show that $f$ and $g$ are functions by using the definition of inverse functions. $f ( x ) = 5 x + 1 , g ( x ) = \frac { x - 1 } { 5 }$
A) $f ( g ( x ) ) = x , g ( f ( x ) ) = x$
B) $f ( g ( x ) ) = x ^ { - 1 } , g ( f ( x ) ) = x$
C) $f ( g ( x ) ) = x , g ( f ( x ) ) = x ^ { - 1 }$
D) $f ( g ( x ) ) = x + 1 , g ( f ( x ) ) = x$
E) $f ( g ( x ) ) = x , g ( f ( x ) ) = x + 1$

Accepted Answer

To show that $f$ and $g$ are inverse functions, we need to demonstrate that $f(g(x)) = x$ and $g(f(x)) = x$. For $f(g(x))$, substitute $g(x) = \frac{x - 1}{5}$ into $f$, yielding $f(g(x)) = 5\left(\frac{x - 1}{5}\right) + 1 = x$. For $g(f(x))$, substitute $f(x) = 5x + 1$ into $g$, yielding $g(f(x)) = \frac{(5x + 1) - 1}{5} = x$. Both compositions return $x$, confirming that $f$ and $g$ are indeed inverse functions.

Question 7

Find the inverse function of the function f given by the set of ordered pairs. $\{ ( 6,2 ) , ( 5,3 ) , ( 4,4 ) , ( 3,5 ) \}$
A) $f ^ { - 1 } = \{ ( 2,6 ) , ( 3,5 ) , ( 4,4 ) , ( 5,3 ) \}$
B) $f ^ { - 1 } = \{ ( 2,6 ) , ( 3,5 ) , ( 4,4 ) , ( 5,5 ) \}$
C) $f ^ { - 1 } = \{ ( 2,6 ) , ( 3,5 ) , ( 4,2 ) , ( 5,3 ) \}$
D) $f ^ { - 1 } = \{ ( 2,6 ) , ( 3,3 ) , ( 4,4 ) , ( 5,3 ) \}$
E) $f ^ { - 1 } = \{ ( 2,2 ) , ( 3,5 ) , ( 4,4 ) , ( 5,3 ) \}$

Accepted Answer

To find the inverse of a function represented by ordered pairs, you swap the positions of each pair's elements. The correct inverse for the given function is $\{ ( 2,6 ) , ( 3,5 ) , ( 4,4 ) , ( 5,3 ) \}$.

Question 8

Sketch the graphs of inverse functions $f ( x ) = x ^ { 5 } - 2 , f ^ { - 1 } ( x ) = \sqrt [ 5 ] { x + 2 }$ in the same coordinate plane and show that the graphs are reflections of each other in the line $y = x$
A)  $\begin{array} { l l r r } 
x & - 1 & 0 & 1 \
f ( x ) & - 3 & - 2 & - 1 \\
x & - 4 & - 2 & - 1 \
f ^ { - 1 } ( x ) & - 1 & 0 & 1
\end{array}$
B)  $\begin{array} { l lrr } 
x & - 1 & 0 & 1 \
f ( x ) & - 3 & - 2 & - 1 \\
x & - 3 & - 2 & -1\
f ^ { - 1 } ( x ) & - 1 & 2 &  1
\end{array}$
C)  $\begin{array} { l r rr } 
x & 0 & 0 & 1 \
f ( x ) & - 3 & - 2 & - 1 \\
x & - 3 & - 2 & -1\
f ^ { - 1 } ( x ) & - 1 & 0 &  1
\end{array}$
D)  $\begin{array} { l l rr } 
x & - 1 & 0 & 1 \
f ( x ) & - 3 & - 2 & - 1 \\
x & - 3 & - 2 & -1\
f ^ { - 1 } ( x ) & - 1 & 0 & 1
\end{array}$
E) 
$\begin{array}{llrr}
x & -1 & 0 & -1 \
f(x) & -3 & -2 & -1 \\
x & -3& 2& -1 \
f^{-1}(x) & -1 & -2 & 1
\end{array}$

Accepted Answer

The answer of Sketch the graphs of inverse functions \(f...

Question 9

Find the inverse function informally $f ( x ) = x - 5$ . Verify that $f \left( f ^ { - 1 } ( x ) \right) = x$ and $f ^ { - 1 } ( f ( x ) ) = x$
A) $f ^ { - 1 } ( x ) = x + 1$ , $f \left( f ^ { - 1 } ( x ) \right) = x$ , $f ^ { - 1 } ( f ( x ) ) = x$
B) $f ^ { - 1 } ( x ) = x + 5$ , $f \left( f ^ { - 1 } ( x ) \right) = x$ , $f ^ { - 1 } ( f ( x ) ) = x$
C) $f ^ { - 1 } ( x ) = x + 5$ , $f \left( f ^ { - 1 } ( x ) \right) = x$ , $f ^ { - 1 } ( f ( x ) ) = x - 1$
D) $f ^ { - 1 } ( x ) = x + 5$ , $f \left( f ^ { - 1 } ( x ) \right) = x + 1$ , $f ^ { - 1 } ( f ( x ) ) = x$
E) $f ^ { - 1 } ( x ) = x + 5$ , $f \left( f ^ { - 1 } ( x ) \right) = x$ , $f ^ { - 1 } ( f ( x ) ) = x ^ { - 1 }$

Accepted Answer

The inverse function of $f(x) = x - 5$ is $f^{-1}(x) = x + 5$ because applying $f^{-1}$ to $f(x)$ or vice versa returns the original input $x$, satisfying the conditions for inverse functions.

Question 10

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 3$ and $g ( x ) = x ^ { 2 }$ to find the value $\left( g ^ { - 1 } \circ f ^ { - 1 } \right) ( - 3 )$
A)1
B)0
C)3
D)4
E)2

Accepted Answer

The answer of Use the functions given by \(f (...

Question 11

Find the inverse function of the function f given by the set of ordered pairs. $\{ ( 1,4 ) , ( 2,5 ) , ( 3,6 ) , ( 4,7 ) \}$
A) $f ^ { - 1 } = \{ ( 4,4 ) , ( 5,2 ) , ( 6,3 ) , ( 7,4 ) \}$
B) $f ^ { - 1 } = \{ ( 4,1 ) , ( 5,5 ) , ( 6,3 ) , ( 7,4 ) \}$
C) $f ^ { - 1 } = \{ ( 4,1 ) , ( 5,2 ) , ( 6,6 ) , ( 7,4 ) \}$
D) $f ^ { - 1 } = \{ ( 4,1 ) , ( 5,2 ) , ( 6,3 ) , ( 7,4 ) \}$
E) $f ^ { - 1 } = \{ ( 4,1 ) , ( 5,2 ) , ( 6,3 ) , ( 7,7 ) \}$

Accepted Answer

To find the inverse of a function represented by ordered pairs, you swap the positions of each element in the pairs. The original function $f$ is given by $\{ ( 1,4 ) , ( 2,5 ) , ( 3,6 ) , ( 4,7 ) \}$, so its inverse $f^{-1}$ is $\{ ( 4,1 ) , ( 5,2 ) , ( 6,3 ) , ( 7,4 ) \}$.

Question 12

Find the inverse function informally $f ( x ) = 2 x$ . Verify that $f \left( f ^ { - 1 } ( x ) \right) = x$ and $f ^ { - 1 } ( f ( x ) ) = x$
A) $f ^ { - 1 } ( x ) = \frac { x } { 2 }$ , $f \left( f ^ { - 1 } ( x ) \right) = x ^ { - 1 }$ , $f ^ { - 1 } ( f ( x ) ) = x$
B) $f ^ { - 1 } ( x ) = \frac { x } { 2 }$ , $f \left( f ^ { - 1 } ( x ) \right) = 2 x$ , $f ^ { - 1 } ( f ( x ) ) = 3 x$
C) $f ^ { - 1 } ( x ) = \frac { x } { 2 }$ , $f \left( f ^ { - 1 } ( x ) \right) = x$ , $f ^ { - 1 } ( f ( x ) ) = 2 x$
D) $f ^ { - 1 } ( x ) = \frac { x } { 2 }$ , $f \left( f ^ { - 1 } ( x ) \right) = 3 x$ , $f ^ { - 1 } ( f ( x ) ) = x$
E) $f ^ { - 1 } ( x ) = \frac { x } { 2 }$ , $f \left( f ^ { - 1 } ( x ) \right) = x$ , $f ^ { - 1 } ( f ( x ) ) = x$

Accepted Answer

The inverse function of $f(x) = 2x$ is $f^{-1}(x) = \frac{x}{2}$ because applying $f$ to $f^{-1}(x)$ or vice versa returns the original input $x$, satisfying the conditions for inverse functions.

Question 13

Use the graph of $f$ to complete the table and to sketch the graph of $f ^ { - 1 }$ $\begin{array}{llllll}
x & 0 & 1 & 2 & 3 & 4\
f^{-1}(x)
\end{array}$ 
A) $\begin{array} { c c c c c c } 
x & 0 & 1 & 2 & 3 & 4 \
f ^ { - 1 } ( x ) & 0 & 4 & - 2 & 2 & 1
\end{array}$ 
B) $\begin{array} { c c c c c c } 
x & 0 & 1 & 2 & 3 & 4 \
f ^ { - 1 } ( x ) & 4 & - 2 & 2 & 0 & 1
\end{array}$ 
C) $\begin{array} { l l l l l l } 
x & 0 & 1 & 2 & 3 & 4 \
f ^ { - 1 } ( x ) & - 2 & 0 & 1 & 2 & 4
\end{array}$ 
D) $\begin{array} { c c c c c c } 
x & 0 & 1 & 2 & 3 & 4 \
f ^ { - 1 } ( x ) & 2 & 4 & - 2 & 0 & 1
\end{array}$ 
E) $\begin{array} { c c c c c c } 
x & 0 & 1 & 2 & 3 & 4 \
f ^ { - 1 } ( x ) & 1 & 2 & - 2 & 4 & 0
\end{array}$

Accepted Answer

The answer of Use the graph of $f$ to complete...

Question 14

Determine whether the function has an inverse function, If it does, find its inverse function. $f ( x ) = 36 + x ^ { 2 } , x \leq 0$
A)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = - \sqrt { x - 36 } , x \geq 36$
B)No, $f$ does not have an inverse function.
C)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = - \sqrt { x + 36 } , x \geq 36$
D)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = \sqrt { x - 36 } , x \geq 36$
E)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = - \sqrt { x - 36 } , x \leq 36$

Accepted Answer

Given $f(x) = 36 + x^2$ for $x \leq 0$, the function is one-to-one and thus has an inverse. To find the inverse, solve for $x$ in terms of $y$, leading to $x = -\sqrt{y - 36}$, which is defined for $y \geq 36$.

Question 15

Use the graph of $f$ to determine whether the function has an inverse function. 
A)Yes, $f$ has an inverse function.
B)No, $f$ does not have an inverse function.

Accepted Answer

The answer of Use the graph of $f$ to determine...

Question 16

A company's profit $P$ for producing $x$ units is given by $P ( x ) = 47 x - 5736$ . Find the inverse function $P ^ { - 1 } ( x )$ and explain what it represents. Describe the domains of $P ( x )$ and $P ^ { - 1 } ( x )$ .
A) $P ^ { - 1 } ( x ) = \frac { x + 5736 } { 47 }$ , $P ^ { - 1 } ( x )$ represents the number of units that must be sold to obtain the profit of $x$ .Domain of $P ( x ) : [ 0,10 )$ , Domain of $P ^ { - 1 } ( x ) : [ - 5736 , \infty )$
B) $P ^ { - 1 } ( x ) = \frac { x + 5736 } { 47 }$ , $P ^ { - 1 } ( x )$ represents the number of units that must be sold to obtain the profit of $x$ .Domain of $P ( x ) : [ 0 , \infty )$ , Domain of $P ^ { - 1 } ( x ) : [ - 5736,5736 )$
C) $P ^ { - 1 } ( x ) = \frac { x + 5736 } { 47 }$ , $P ^ { - 1 } ( x )$ represents the number of units that must be sold to obtain the profit of $x$ .Domain of $P ( x ) : [ 0 , \infty )$ , Domain of $P ^ { - 1 } ( x ) : [ - 5736 , \infty )$
D) $P ^ { - 1 } ( x ) = \frac { x + 5736 } { 47 }$ , $P ^ { - 1 } ( x )$ represents the number of units that must be sold to obtain the profit of $x$ .Domain of $P ( x ) : [ 1 , \infty )$ , Domain of $P ^ { - 1 } ( x ) : [ - 5736 , \infty )$
E) $P ^ { - 1 } ( x ) = \frac { x + 5736 } { 47 }$ , $P ^ { - 1 } ( x )$ represents the number of units that must be sold to obtain the profit of $x$ .Domain of $P ( x ) : [ 0 , \infty )$ , Domain of $P ^ { - 1 } ( x ) : [ - 5736,47 )$

Accepted Answer

The inverse function $P^{-1}(x) = \frac{x + 5736}{47}$ is found by solving the equation $P(x) = x$ for $x$, which represents the number of units needed to achieve a profit of $x$. The domain of $P(x)$ is $[0, \infty)$ because a company cannot produce a negative number of units. The domain of $P^{-1}(x)$ is $[-5736, \infty)$ because the profit can be negative (indicating a loss) up to $-5736$ when no units are sold, and there's no upper limit to the profit.

Question 17

Sketch the graphs of inverse functions $f ( x ) = \frac { x } { 3 } , f ^ { - 1 } ( x ) = 3 x$ in the same coordinate plane and show that the graphs are reflections of each other in the line $y = x$
A)  $\begin{array} { l l l l } 
x & - 3 & 0 & 3 \
f ( x ) & - 1 & 0 & 1
\end{array}$ $\begin{array} { l l l l } 
x & - 1 & 0 & 1 \
f ^ { - 1 } ( x ) & - 3 & 0 & 3
\end{array}$
B)  $\begin{array} { l l l l } 
x & - 2 & 0 & 3 \
f ( x ) & - 1 & 0 & 1
\end{array}$ $\begin{array} { l l l l } 
x & - 1 & 0 & 1 \
f ^ { - 1 } ( x ) & - 3 & 0 & 3
\end{array}$
C)  $\begin{array} { l l l l } 
x & - 3 & 0 & 3 \
f ( x ) & - 1 & 0 & 1
\end{array}$ $\begin{array} { l l l l } 
x & - 1 & 0 & 1 \
f ^ { - 1 } ( x ) & - 3 & 2 & 3
\end{array}$
D)  $\begin{array} { l l l l } 
x & - 3 & 0 & 0 \
f ( x ) & - 1 & 0 & 1
\end{array}$ $\begin{array} { l l l l } 
x & - 1 & 0 & 1 \
f ^ { - 1 } ( x ) & - 3 & 0 & 3
\end{array}$
E)  $\begin{array} { l l l l } 
x & - 3 & 0 & 3 \
f ( x ) & - 1 & 0 & 1
\end{array}$ $\begin{array} { l l l l } 
x & - 1 & 0 & 1 \
f ^ { - 1 } ( x ) & - 2 & 0 & 3
\end{array}$

Accepted Answer

The answer of Sketch the graphs of inverse functions \(f...

Question 18

Determine whether the function has an inverse function, If it does, find its inverse function. $f ( x ) = x ^ { 4 }$
A)No, $f$ does not have an inverse function.
B)Yes, $f$ has an inverse function, $y = x ^ { 4 }$
C)Yes, $f$ has an inverse function, $y = - x ^ { - 4 }$
D)Yes, $f$ has an inverse function, $y = x ^ { - 4 }$
E)Yes, $f$ has an inverse function, $y = - x ^ { 4 }$

Accepted Answer

The function $f(x) = x^4$ does not have an inverse function because it is not one-to-one; for any given y-value, there are two x-values (one positive and one negative) that produce the same result when raised to the fourth power, violating the requirement for a function to have an inverse.

Question 19

Use the graph of $f$ to determine whether the function has an inverse function. 
A)No, $f$ does not have an inverse function.
B)Yes, $f$ has an inverse function.

Accepted Answer

The answer of Use the graph of $f$ to determine...

Question 20

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 3$ and $g ( x ) = x ^ { 2 }$ to find the value $\left( g ^ { - 1 } \circ g ^ { - 1 } \right) ( - 4 )$
A)16
B)-16
C)32
D)The value does not exist.
E)-32

Accepted Answer

The function $g(x) = x^2$ is not one-to-one, so it does not have an inverse over all real numbers. Therefore, $g^{-1}$ does not exist for all values, including -4.

Quiz 5: Exponential and Logarithmic Functions

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 3$ and $g ( x ) = x ^ { 2 }$ to find the value $\left( f ^ { - 1 } \circ g ^ { - 1 } \right) ( 1 )$

Show that $f$ and $g$ are functions by using the definition of inverse functions. $f ( x ) = \frac { 1 } { x } , g ( x ) = \frac { 1 } { x }$

Use the functions given by $f ( x ) = x + 4$ and $g ( x ) = 2 x - 6$ to find the composition of functions $g ^ { - 1 } \circ f ^ { - 1 }$

Determine whether the function has an inverse function, If it does, find its inverse function. $f ( x ) = | x - 2 | , x \leq 2$

Use the functions given by $f ( x ) = x + 4$ and $g ( x ) = 2 x - 6$ to find the composition of functions $(g\circ f) ^{-1}$

Show that $f$ and $g$ are functions by using the definition of inverse functions. $f ( x ) = 5 x + 1 , g ( x ) = \frac { x - 1 } { 5 }$

Find the inverse function of the function f given by the set of ordered pairs. $\{ ( 6,2 ) , ( 5,3 ) , ( 4,4 ) , ( 3,5 ) \}$

Sketch the graphs of inverse functions $f ( x ) = x ^ { 5 } - 2 , f ^ { - 1 } ( x ) = \sqrt [ 5 ] { x + 2 }$ in the same coordinate plane and show that the graphs are reflections of each other in the line $y = x$

Find the inverse function informally $f ( x ) = x - 5$ . Verify that $f \left( f ^ { - 1 } ( x ) \right) = x$ and $f ^ { - 1 } ( f ( x ) ) = x$

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 3$ and $g ( x ) = x ^ { 2 }$ to find the value $\left( g ^ { - 1 } \circ f ^ { - 1 } \right) ( - 3 )$

Find the inverse function of the function f given by the set of ordered pairs. $\{ ( 1,4 ) , ( 2,5 ) , ( 3,6 ) , ( 4,7 ) \}$

Find the inverse function informally $f ( x ) = 2 x$ . Verify that $f \left( f ^ { - 1 } ( x ) \right) = x$ and $f ^ { - 1 } ( f ( x ) ) = x$

Determine whether the function has an inverse function, If it does, find its inverse function. $f ( x ) = 36 + x ^ { 2 } , x \leq 0$

Use the graph of $f$ to determine whether the function has an inverse function.

A company's profit $P$ for producing $x$ units is given by $P ( x ) = 47 x - 5736$ . Find the inverse function $P ^ { - 1 } ( x )$ and explain what it represents. Describe the domains of $P ( x )$ and $P ^ { - 1 } ( x )$ .

Sketch the graphs of inverse functions $f ( x ) = \frac { x } { 3 } , f ^ { - 1 } ( x ) = 3 x$ in the same coordinate plane and show that the graphs are reflections of each other in the line $y = x$

Determine whether the function has an inverse function, If it does, find its inverse function. $f ( x ) = x ^ { 4 }$

Use the graph of $f$ to determine whether the function has an inverse function.

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 3$ and $g ( x ) = x ^ { 2 }$ to find the value $\left( g ^ { - 1 } \circ g ^ { - 1 } \right) ( - 4 )$

Quiz 5: Exponential and Logarithmic Functions

Use the functions given by f(x) =18x−3f ( x ) = \frac { 1 } { 8 } x - 3f(x) =81​x−3 and g(x) =x2g ( x ) = x ^ { 2 }g(x) =x2 to find the value (f−1∘g−1) (1) \left( f ^ { - 1 } \circ g ^ { - 1 } \right) ( 1 ) (f−1∘g−1) (1)

Show that fff and ggg are functions by using the definition of inverse functions. f(x) =1x,g(x) =1xf ( x ) = \frac { 1 } { x } , g ( x ) = \frac { 1 } { x }f(x) =x1​,g(x) =x1​

Use the functions given by f(x) =x+4f ( x ) = x + 4f(x) =x+4 and g(x) =2x−6g ( x ) = 2 x - 6g(x) =2x−6 to find the composition of functions g−1∘f−1g ^ { - 1 } \circ f ^ { - 1 }g−1∘f−1

Determine whether the function has an inverse function, If it does, find its inverse function. f(x) =∣x−2∣,x≤2f ( x ) = | x - 2 | , x \leq 2f(x) =∣x−2∣,x≤2

Use the functions given by f(x) =x+4f ( x ) = x + 4f(x) =x+4 and g(x) =2x−6g ( x ) = 2 x - 6g(x) =2x−6 to find the composition of functions (g∘f) −1(g\circ f) ^{-1}(g∘f) −1

Show that fff and ggg are functions by using the definition of inverse functions. f(x) =5x+1,g(x) =x−15f ( x ) = 5 x + 1 , g ( x ) = \frac { x - 1 } { 5 }f(x) =5x+1,g(x) =5x−1​

Find the inverse function of the function f given by the set of ordered pairs. {(6,2) ,(5,3) ,(4,4) ,(3,5) }\{ ( 6,2 ) , ( 5,3 ) , ( 4,4 ) , ( 3,5 ) \}{(6,2) ,(5,3) ,(4,4) ,(3,5) }

Sketch the graphs of inverse functions f(x) =x5−2,f−1(x) =x+25f ( x ) = x ^ { 5 } - 2 , f ^ { - 1 } ( x ) = \sqrt [ 5 ] { x + 2 }f(x) =x5−2,f−1(x) =5x+2​ in the same coordinate plane and show that the graphs are reflections of each other in the line y=xy = xy=x

Find the inverse function informally f(x) =x−5f ( x ) = x - 5f(x) =x−5 . Verify that f(f−1(x) ) =xf \left( f ^ { - 1 } ( x ) \right) = xf(f−1(x) ) =x and f−1(f(x) ) =xf ^ { - 1 } ( f ( x ) ) = xf−1(f(x) ) =x

Use the functions given by f(x) =18x−3f ( x ) = \frac { 1 } { 8 } x - 3f(x) =81​x−3 and g(x) =x2g ( x ) = x ^ { 2 }g(x) =x2 to find the value (g−1∘f−1) (−3) \left( g ^ { - 1 } \circ f ^ { - 1 } \right) ( - 3 ) (g−1∘f−1) (−3)

Find the inverse function of the function f given by the set of ordered pairs. {(1,4) ,(2,5) ,(3,6) ,(4,7) }\{ ( 1,4 ) , ( 2,5 ) , ( 3,6 ) , ( 4,7 ) \}{(1,4) ,(2,5) ,(3,6) ,(4,7) }

Find the inverse function informally f(x) =2xf ( x ) = 2 xf(x) =2x . Verify that f(f−1(x) ) =xf \left( f ^ { - 1 } ( x ) \right) = xf(f−1(x) ) =x and f−1(f(x) ) =xf ^ { - 1 } ( f ( x ) ) = xf−1(f(x) ) =x

Use the graph of fff to complete the table and to sketch the graph of f−1f ^ { - 1 }f−1 x01234f−1(x) \begin{array}{llllll}x & 0 & 1 & 2 & 3 & 4\\f^{-1}(x) \end{array}xf−1(x) ​01234

Determine whether the function has an inverse function, If it does, find its inverse function. f(x) =36+x2,x≤0f ( x ) = 36 + x ^ { 2 } , x \leq 0f(x) =36+x2,x≤0

Use the graph of fff to determine whether the function has an inverse function.

A company's profit PPP for producing xxx units is given by P(x) =47x−5736P ( x ) = 47 x - 5736P(x) =47x−5736 . Find the inverse function P−1(x) P ^ { - 1 } ( x ) P−1(x) and explain what it represents. Describe the domains of P(x) P ( x ) P(x) and P−1(x) P ^ { - 1 } ( x ) P−1(x) .

Sketch the graphs of inverse functions f(x) =x3,f−1(x) =3xf ( x ) = \frac { x } { 3 } , f ^ { - 1 } ( x ) = 3 xf(x) =3x​,f−1(x) =3x in the same coordinate plane and show that the graphs are reflections of each other in the line y=xy = xy=x

Determine whether the function has an inverse function, If it does, find its inverse function. f(x) =x4f ( x ) = x ^ { 4 }f(x) =x4

Use the graph of fff to determine whether the function has an inverse function.

Use the functions given by f(x) =18x−3f ( x ) = \frac { 1 } { 8 } x - 3f(x) =81​x−3 and g(x) =x2g ( x ) = x ^ { 2 }g(x) =x2 to find the value (g−1∘g−1) (−4) \left( g ^ { - 1 } \circ g ^ { - 1 } \right) ( - 4 ) (g−1∘g−1) (−4)

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 3$ and $g ( x ) = x ^ { 2 }$ to find the value $\left( f ^ { - 1 } \circ g ^ { - 1 } \right) ( 1 )$

Show that $f$ and $g$ are functions by using the definition of inverse functions. $f ( x ) = \frac { 1 } { x } , g ( x ) = \frac { 1 } { x }$

Use the functions given by $f ( x ) = x + 4$ and $g ( x ) = 2 x - 6$ to find the composition of functions $g ^ { - 1 } \circ f ^ { - 1 }$

Determine whether the function has an inverse function, If it does, find its inverse function. $f ( x ) = | x - 2 | , x \leq 2$

Use the functions given by $f ( x ) = x + 4$ and $g ( x ) = 2 x - 6$ to find the composition of functions $(g\circ f) ^{-1}$

Show that $f$ and $g$ are functions by using the definition of inverse functions. $f ( x ) = 5 x + 1 , g ( x ) = \frac { x - 1 } { 5 }$

Find the inverse function of the function f given by the set of ordered pairs. $\{ ( 6,2 ) , ( 5,3 ) , ( 4,4 ) , ( 3,5 ) \}$

Sketch the graphs of inverse functions $f ( x ) = x ^ { 5 } - 2 , f ^ { - 1 } ( x ) = \sqrt [ 5 ] { x + 2 }$ in the same coordinate plane and show that the graphs are reflections of each other in the line $y = x$

Find the inverse function informally $f ( x ) = x - 5$ . Verify that $f \left( f ^ { - 1 } ( x ) \right) = x$ and $f ^ { - 1 } ( f ( x ) ) = x$

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 3$ and $g ( x ) = x ^ { 2 }$ to find the value $\left( g ^ { - 1 } \circ f ^ { - 1 } \right) ( - 3 )$

Find the inverse function of the function f given by the set of ordered pairs. $\{ ( 1,4 ) , ( 2,5 ) , ( 3,6 ) , ( 4,7 ) \}$

Find the inverse function informally $f ( x ) = 2 x$ . Verify that $f \left( f ^ { - 1 } ( x ) \right) = x$ and $f ^ { - 1 } ( f ( x ) ) = x$

Determine whether the function has an inverse function, If it does, find its inverse function. $f ( x ) = 36 + x ^ { 2 } , x \leq 0$

Use the graph of $f$ to determine whether the function has an inverse function.

A company's profit $P$ for producing $x$ units is given by $P ( x ) = 47 x - 5736$ . Find the inverse function $P ^ { - 1 } ( x )$ and explain what it represents. Describe the domains of $P ( x )$ and $P ^ { - 1 } ( x )$ .

Sketch the graphs of inverse functions $f ( x ) = \frac { x } { 3 } , f ^ { - 1 } ( x ) = 3 x$ in the same coordinate plane and show that the graphs are reflections of each other in the line $y = x$

Determine whether the function has an inverse function, If it does, find its inverse function. $f ( x ) = x ^ { 4 }$

Use the graph of $f$ to determine whether the function has an inverse function.

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 3$ and $g ( x ) = x ^ { 2 }$ to find the value $\left( g ^ { - 1 } \circ g ^ { - 1 } \right) ( - 4 )$