Question 1

The supply function for a product is $p ( x ) = \frac { 1 } { 3 } x ^ { 2 } + 50$ where  x  is the number of thousands of units a manufacturer will supply if the price is  p(x)  dollars. Find the inverse of this function.
A) $p ^ { - 1 } ( x ) = \frac { 1 } { 3 } \sqrt { x } + 50$
B) $\mathrm { p } ^ { - 1 } ( \mathrm { x } ) = 3 \sqrt { \mathrm { x } } - 50$
C) $\mathrm { p } ^ { - 1 } ( \mathrm { x } ) = 3 \sqrt { ( \mathrm { x } - 50 ) }$
D) $p ^ { - 1 } ( x ) = \sqrt { 3 ( x - 50 ) }$

Accepted Answer

To find the inverse function, we first replace $p(x)$ with $y$, giving us $y = \frac{1}{3}x^2 + 50$. Then, we solve for $x$ in terms of $y$. Swapping $x$ and $y$ gives us $x = \frac{1}{3}y^2 + 50$, but we need to solve for $y$, so we rearrange to get $y^2 = 3(x - 50)$. Taking the square root of both sides gives $y = \sqrt{3(x - 50)}$, which corresponds to the inverse function $p^{-1}(x) = \sqrt{3(x - 50)}$.

Question 2

The surface area of a sphere is $f ( x ) = 4 \pi x ^ { 2 }$ square inches, where  x  is the radius of the sphere in inches. Find the inverse of this function.
A) $f ^ { - 1 } ( x ) = \frac { \sqrt { x } } { 2 \pi }$
B) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 2 } \sqrt { \frac { \mathrm { x } } { \pi } }$
C) $f ^ { - 1 } ( x ) = 2 \sqrt { \frac { \pi } { x } }$
D) $f ^ { - 1 } ( x ) = 2 \sqrt { \pi x }$

Accepted Answer

To find the inverse function, you swap $x$ and $y$ and solve for $y$. Starting with $y = 4\pi x^2$, swap to get $x = 4\pi y^2$, and solve for $y$: $y = \frac{1}{2}\sqrt{\frac{x}{\pi}}$.

Question 3

Provide an appropriate response.
-Why does a linear function always have an inverse?

Accepted Answer

The answer of Provide an appropriate response.
-Why does a linear...

Question 4

Provide an appropriate response.
-Why does the horizontal line test (checking if a function is one-to-one) work?

Accepted Answer

The answer of Provide an appropriate response.
-Why does the horizontal...

Question 5

Encode the message BEWARE OF DOG using the one-to-one function defined by $f ( x ) = x ^ { 3 } - 1$ - Use the alphabet coding method  (A=1, B=2, C=3, ...) . Give the inverse function that the decoder would need when the message is received.

Accepted Answer

The answer of Encode the message BEWARE OF DOG using...

Question 6

A size 8 dress in Country  C  is size 48 in Country D. A function that converts dress sizes in Country C to those in Country D is  f(x)=2( x +16) . Find a formula for the inverse of this function.
A) $f ^ { - 1 } ( x ) = \frac { x - 16 } { 2 }$
B) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \mathrm { x } - 16$
C) $f ^ { - 1 } ( x ) = \frac { x } { 2 } + 16$
D) $f ^ { - 1 } ( x ) = \frac { x } { 2 } - 16$

Accepted Answer

The answer of A size 8 dress in Country ...

Question 7

The graph of the function y $y = f ( x ) \text { is given. On the same axes, sketch the graph of } f ^ { - 1 } ( x )$ . Use a dashed line for the inverse
function.
-A size 4 dress in Country C is size 42 in Country D. A function that converts dress sizes in Country $C$ to those in Country $D$ is $f ( x ) = x + 38$. Find a formula for the inverse of this function.
A) $f ^ { - 1 } ( x ) = x + 38$
B) $f ^ { - 1 } ( x ) = \frac { x } { - 38 }$
C) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { \mathrm { x } } { 38 }$
D) $f ^ { - 1 } ( x ) = x - 38$

Accepted Answer

To find the inverse function $f^{-1}(x)$, you swap $x$ and $y$ in the original function and solve for $y$. Starting with $y = x + 38$, swapping gives $x = y + 38$, and solving for $y$ gives $y = x - 38$, which corresponds to $f^{-1}(x) = x - 38$.

Question 8

A size 44 dress in Country  C  is size 2 in Country D. A function that converts dress sizes in Country  C  to those in Country D is : $f ( x ) = \frac { x } { 2 } - 20$ . Find a formula for the inverse of this function.
A) $f ^ { - 1 } ( x ) = x + 20$
B) $f ^ { - 1 } ( x ) = 2 x + 20$
C) $f ^ { - 1 } ( x ) = 2 ( x - 20 )$
D) $f ^ { - 1 } ( x ) = 2 ( x + 20 )$

Accepted Answer

The answer of A size 44 dress in Country ...

Question 9

The function defined by  f(x)=x+2  was used to encode the following message:
15 3 22 10 9 7 16 11 23 21
using the alphabet coding method  (A=1, B=2, C=3, ...) . Find the inverse and decode the message.

Accepted Answer

The answer of The function defined by  f(x)=x+2 ...

Question 10

An organization determines that the cost per person of chartering a bus is given by the formula $C ( x ) = \frac { 200 + 7 x } { x }$
where  x  is the number of people in the group and  C(x)  is in dollars. Find a formula for the inverse of this functio
A) $C ^ { - 1 } ( x ) = \frac { 200 } { x + 7 }$
B) $C ^ { - 1 } ( x ) = \frac { 7 } { x - 200 }$
C) $C ^ { - 1 } ( x ) = \frac { 200 } { x - 7 }$
D) $C ^ { - 1 } ( x ) = \frac { 200 + x } { 7 }$

Accepted Answer

The answer of An organization determines that the cost per...

Question 11

On the day when Julie left for her vacation, the function that converted U.S. dollars to British pounds was  f(x)=   0.54521 x , where  x  is the number of U.S. dollars and  f(x)  is the number of British pounds. Find the inverse function.
A) $f ^ { - 1 } ( x ) = \frac { x } { 0.54521 }$
B) $f ^ { - 1 } ( x ) = x - 0.54521$
C) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 0.54521 } { \mathrm { x } }$
D) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 0.54521 \mathrm { x } }$

Accepted Answer

The inverse function $f^{-1}(x)$ is found by solving for $x$ in terms of $f(x)$, which leads to $x = \frac{f(x)}{0.54521}$, thus $f^{-1}(x) = \frac{x}{0.54521}$.

Question 12

The function that models the consumption of a certain medication over the years  2000-2010  is  f(x)=115.21x+224.10 , where x  is the number of years after 2000 and  y  is measured in grams per 100,000 people. Find the inverse of this function.
A) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { \mathrm { x } } { 115.21 } - 224.10$
B) $f ^ { - 1 } ( x ) = \frac { 1 } { 115.21 x + 224.10 }$
C) $f ^ { - 1 } ( x ) = 115.21 ( x - 224.10 )$
D) $f ^ { - 1 } ( x ) = \frac { x - 224.10 } { 115.21 }$

Accepted Answer

To find the inverse function, you swap $x$ and $y$ and solve for $y$. Starting with $y = 115.21x + 224.10$, swapping gives $x = 115.21y + 224.10$. Solving for $y$ gives $y = \frac{x - 224.10}{115.21}$, which matches choice D.

Question 13

Determine whether the function is one-to-one.
-The function that pairs students' ID numbers with their GPAs.
A) Yes
B) No

Accepted Answer

The answer of Determine whether the function is one-to-one.
-The function...

Question 14

The volume of a cylinder whose height is equal to twice its radius is $\mathrm { f } ( \mathrm { x } ) = 2 \pi \mathrm { x } ^ { 3 } \mathrm {~cm} ^ { 3 }$ , where  x  is the radius of the cylinder in  cm . Find the inverse of this function.
A) $f ^ { - 1 } ( x ) = 2 \pi \sqrt [ 3 ] { x }$
B) $f ^ { - 1 } ( x ) = \frac { 1 } { 2 \pi x ^ { 3 } }$
C) $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { \frac { x } { 2 \pi } }$
D) $f ^ { - 1 } ( x ) = \frac { \sqrt [ 3 ] { x } } { 2 \pi }$

Accepted Answer

To find the inverse function, solve for $x$ in terms of $f(x)$. Given $f(x) = 2\pi x^3$, solving for $x$ gives $x = \sqrt[3]{\frac{f(x)}{2\pi}}$, which corresponds to option C when replacing $f(x)$ with $x$ for the inverse function notation.

Question 15

Provide an appropriate response.
-Your classmate missed the lecture on how to find the inverse of a one-to-one function. When your classmate
calls you on the phone for an explanation, what will you say to explain the process?

Accepted Answer

The answer of Provide an appropriate response.
-Your classmate missed the...

Question 16

Determine whether the function is one-to-one.
-The function that pairs the temperature at 3:00 pm in your city in degrees Fahrenheit with the temperature in degrees Celsius.
A) Yes
B) No

Accepted Answer

The function is one-to-one because for each temperature in Fahrenheit at 3:00 pm, there is a unique corresponding temperature in Celsius, following the linear conversion formula $C = \frac{5}{9}(F - 32)$. This ensures a one-to-one relationship.

Question 17

The graph of the function y $y = f ( x ) \text { is given. On the same axes, sketch the graph of } f ^ { - 1 } ( x )$ . Use a dashed line for the inverse
function.
-
A)

B)

C)

D)

Accepted Answer

The answer of The graph of the function y \(y...

Question 18

A size 12 dress in Country  C  is size  -22  in Country D. A function that converts dress sizes in Country C to those in Country D is  f(x)= x -34 . Find a formula for the inverse of this function.
A) $f ^ { - 1 } ( x ) = \frac { x } { 34 }$
B) $f ^ { - 1 } ( x ) = x + 34$
C) $f ^ { - 1 } ( x ) = x - 34$
D) $f ^ { - 1 } ( x ) = \frac { x } { - 34 }$

Accepted Answer

To find the inverse function, you swap $x$ and $y$ and solve for $y$. If $f(x) = x - 34$, then for the inverse, $x = y - 34$, which simplifies to $y = x + 34$.

Question 19

The graph of the function y $y = f ( x ) \text { is given. On the same axes, sketch the graph of } f ^ { - 1 } ( x )$ . Use a dashed line for the inverse
function.
-

A)

B)

C)

D)

Accepted Answer

The answer of The graph of the function y \(y...

Question 20

$32 ^ { \circ }$ Fahrenheit  = $0 ^ { \circ }$ Celsius. A function that converts temperatures in Celsius to those in Fahrenheit is $f ( x ) = \frac { 9 } { 5 } x + 32$ Fahrenheit. Find a formula for the inverse of this function.
A) $f ^ { - 1 } ( x ) = x + 32$
B) $f ^ { - 1 } ( x ) = \frac { 5 } { 9 } ( x - 32 )$
C) $f ^ { - 1 } ( x ) = \frac { 5 } { 9 } ( x + 32 )$
D) $f ^ { - 1 } ( x ) = \frac { 9 } { 5 } x + 32$

Accepted Answer

To find the inverse function, $f^{-1}(x)$, we swap $x$ and $y$ in the original function $y = \frac{9}{5}x + 32$ and solve for $y$. This gives $x = \frac{9}{5}y + 32$, which when solved for $y$, gives $y = \frac{5}{9}(x - 32)$, corresponding to choice B.

Quiz 4: Additional Topics With Functions

The supply function for a product is $p ( x ) = \frac { 1 } { 3 } x ^ { 2 } + 50$ where x is the number of thousands of units a manufacturer will supply if the price is p(x) dollars. Find the inverse of this function.

The surface area of a sphere is $f ( x ) = 4 \pi x ^ { 2 }$ square inches, where x is the radius of the sphere in inches. Find the inverse of this function.

Provide an appropriate response. -Why does a linear function always have an inverse?

Provide an appropriate response. -Why does the horizontal line test (checking if a function is one-to-one) work?

Encode the message BEWARE OF DOG using the one-to-one function defined by $f ( x ) = x ^ { 3 } - 1$ - Use the alphabet coding method (A=1, B=2, C=3, ...) . Give the inverse function that the decoder would need when the message is received.

A size 8 dress in Country C is size 48 in Country D. A function that converts dress sizes in Country C to those in Country D is f(x) =2( x +16) . Find a formula for the inverse of this function.

A size 44 dress in Country C is size 2 in Country D. A function that converts dress sizes in Country C to those in Country D is : $f ( x ) = \frac { x } { 2 } - 20$ . Find a formula for the inverse of this function.

The function defined by f(x)=x+2 was used to encode the following message: 15 3 22 10 9 7 16 11 23 21 using the alphabet coding method (A=1, B=2, C=3, ...) . Find the inverse and decode the message.

An organization determines that the cost per person of chartering a bus is given by the formula $C ( x ) = \frac { 200 + 7 x } { x }$ where x is the number of people in the group and C(x) is in dollars. Find a formula for the inverse of this functio

On the day when Julie left for her vacation, the function that converted U.S. dollars to British pounds was f(x) = 0.54521 x , where x is the number of U.S. dollars and f(x) is the number of British pounds. Find the inverse function.

The function that models the consumption of a certain medication over the years 2000-2010 is f(x) =115.21x+224.10 , where x is the number of years after 2000 and y is measured in grams per 100,000 people. Find the inverse of this function.

Determine whether the function is one-to-one. -The function that pairs students' ID numbers with their GPAs.

The volume of a cylinder whose height is equal to twice its radius is $\mathrm { f } ( \mathrm { x } ) = 2 \pi \mathrm { x } ^ { 3 } \mathrm {~cm} ^ { 3 }$ , where x is the radius of the cylinder in cm . Find the inverse of this function.

Provide an appropriate response. -Your classmate missed the lecture on how to find the inverse of a one-to-one function. When your classmate calls you on the phone for an explanation, what will you say to explain the process?

Determine whether the function is one-to-one. -The function that pairs the temperature at 3:00 pm in your city in degrees Fahrenheit with the temperature in degrees Celsius.

A size 12 dress in Country C is size -22 in Country D. A function that converts dress sizes in Country C to those in Country D is f(x) = x -34 . Find a formula for the inverse of this function.

$32 ^ { \circ }$ Fahrenheit = $0 ^ { \circ }$ Celsius. A function that converts temperatures in Celsius to those in Fahrenheit is $f ( x ) = \frac { 9 } { 5 } x + 32$ Fahrenheit. Find a formula for the inverse of this function.