Inverse Functions - Algebra II

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Question

$f(x) = x^{3} -5$

Which one of the following functions represents the inverse of $f\left ( x \right )?$

A)

B) $\sqrt[3]{x^{3}-5}$

C) $\sqrt[3]{x+5}$

D) $x^{3 } + 5$

E) $x^{3} - 5$

Answer

Given $y = x^{3} - 5$

Hence $x^{3}=y+5$

Interchanging with we get:

$y^{3}=x+5$

Solving for results in $y=\sqrt[3]{x+5}$ .

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Question

$f(x)=\frac{x+3}{x-2}$

Which of the following represents $f^{-1}(x)$ ?

Answer

The question is asking for the inverse function. To find the inverse, first switch input and output -- which is usually easiest if you use notation instead of . Then, solve for .

$f(x)=\frac{x+3}{x-2}$

$y=\frac{x+3}{x-2}$

Here's where we switch:

$x=\frac{y+3}{y-2}$

To solve for , we first have to get it out of the denominator. We do that by multiplying both sides by .

Distribute:

Get all the terms on the same side of the equation:

Factor out a :

Divide by :

$y=\frac{2x+3}{x-1}$

This is our inverse function!

$f^{-1}(x)=\frac{2x+3}{x-1}$

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Question

Find the inverse of .

Answer

To create the inverse, switch x and y making the solution x=3y+3.

y must be isolated to finish the problem.

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Question

Please find the inverse of the following function.

Answer

In order to find the inverse function, we must swap and and then solve for .

Becomes

Now we need to solve for :

$x+\underset{+9}{0}=4y-\underset{+9}{9}$

Finally, we need to divide each side by 4.

$\frac{x+9}{4}=\frac{4y}{4}$

This gives us our inverse function:

$y = \frac{x+9}{4}$

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Question

What is the inverse of the following function?

Answer

Let's say that the function takes the input and yields the output . In math terms:

So, the inverse function needs to take the input and yield the output :

$f^{-1}(b)=a$

So, to answer this question, we need to flip the inputs and outputs for . We do this by replacing with (or a dummy variable; I used ) and with . Then we solve for to get our inverse function:

Now we solve for by subtracting from both sides, taking the cube root, and then adding :

$y=\sqrt[3]{x-3}+4$

is our inverse function, $f^{-1}(x)$

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Question

$g^{-1}(2)=6$

$f^{-1}(-2)=4$

$f^{-1}(9)=2$

What is $f\circ g\circ f^{-1}(6)$ ?

Answer

The question is essentially asking this: take $f^{-1}(6)$ say that equals , then take , then whatever that equals, say , take . So, we start with $f^{-1}(6)$ ; we know that , so if we flip that around we know $f^{-1}(6)=3$ . Now we have to take , but we know that is . Now we have to take , but we don't have that in our table; we do have $f^{-1}(-2)=4$ , though, and if we flip it around, we get , which is our answer.

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Question

$g^{-1}(2)=6$

$f^{-1}(-2)=4$

$f^{-1}(9)=2$

What is $g\circ f\circ g^{-1}(4)$ ?

Answer

Our question is asking "What is of of inverse?" First we find the inverse of . Looking at the question, we see ; if we flip that around, we get $g^{-1}(4)=3$ . Now we need to find what is; that is an easy one, as it is directly provided: . Now we need to find . Again, this isn't given, but what is given is $g^{-1}(2)=6$ , so , and that is our answer.

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Question

Over which line do you flip a function when finding its inverse?

Answer

To find the inverse of a function, you need to change all of the values to values and all the values to values. If you flip a function over the line , then you are changing all the values to values and all the values to values, giving you the inverse of your function.

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Question

Find the inverse of this function:

$f(x)=e^{x^2-3}+\pi$

Answer

To find the inverse of a function, we need to switch all the inputs ( variables) for all the outputs ( variables or variables), so if we just switch all the variables to variables and all the variables to variables and solve for , then will be our inverse function.

$f(x)=e^{x^2-3}+\pi$

turns into the following once the variables are switched:

$x=e^{y^2-3}+\pi$

the first thing we do is subtract $\pi$ from each side; then, we take the natural log of each side. This gives us

$ln(x-\pi)=y^2-3$

Then we just add three to each side and take the square root of each side, making sure we have both the positive and negative roots.

$\pm\sqrt{ln(x-\pi)+3}=y$

This is the inverse function of the function with which we were provided.

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Question

What is the inverse of $y=\frac{2}{5}x -3$ ?

Answer

Interchange the and variables and solve for .

$x=\frac{2}{5}y -3$

$x+3=\frac{2}{5}y$

$y=\frac{5}{2}x+\frac{15}{2}$

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Question

Inverse Functions

Given the function below, find its inverse:

Answer

When finding the inverse go through the following steps:

Replace f(x) with y:

Swap the x and y variables

Solve for y:

add 5 to both sides

divide everything by 3

$f^{^{-1}}(x)=\frac{1}{3}x+\frac{5}{3}$ simplify and express as an inverse using $f^{-1}(x)$

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Question

Find the inverse of .

Answer

To find the inverse of a function, swap the x and y variable and solve for y.

The new expression after the swap is

Now solve for y.

$y^2=\frac{x-7}{3}$

$y=\pm\sqrt{\frac{x-7}{3}}$

This y actually represents the inverse of the original y.

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Question

What is the inverse of the following function?

$f(x)=-2+\sqrt{x-1}$

Answer

To find the inverse of a function, switch the x and y in the original function and then solve for y.

$f(x)=-2+\sqrt{x-1}$

$y=-2+\sqrt{x-1}$

$x=-2+\sqrt{y-1}$

$x+2=\sqrt{y-1}$

$(x+2)^{2}=(\sqrt{y-1})^{2}$

$(x+2)^{2}=y-1$

$y=(x+2)^{2}+1$

$y=x^{2}+4x+5$

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Question

Find the inverse of .

Answer

In order to find the inverse, interchange the and variables.

Solve for . Add eight on both sides.

Divide by three on both sides.

$\frac{x+8}{3}=\frac{3y}{3}$

Simplify the left and right side.

The answer is: $y= \frac{1}{3}x+\frac{8}{3}$

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Question

Find the inverse of:

Answer

Interchange the and variables in the equation.

Solve for . Subtract two on both sides.

Distribute the right side of the equation to eliminate the parentheses.

Subtract four on both sides.

Divide by negative two on both sides.

$\frac{x-6}{-2}=y$

Rewrite this in slope-intercept form.

The answer is: $y= -\frac{1}{2}x + 3$

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Question

If , find $f^{-1}(x)$ .

Answer

The notation $f^{-1}(x)$ denotes the inverse of the function .

Rewrite using as a replacement of .

Interchange and and solve for .

Divide by four on both sides.

$\frac{x}{4}=y^2$

Square root both sides to eliminate the squared term.

$y=\pm \sqrt{\frac{x}{4}} = \pm\frac{ \sqrt x}{2}$

The answer is: $y=\pm\frac{ \sqrt x}{2}$

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Question

Choose the inverse of $y=\frac{x-3}{2}$ .

Answer

To find the inverse of a linear function, switch the variables and solve for y.

$y=\frac{x-3}{2}$

Switch the variables:

$x=\frac{y-3}{2}$

Multiply both sides by 2:

Add 3 to both sides to isolate y:

$\mathbf{2x+3=y}$

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Question

Find the inverse of

Answer

Combine the x terms:

Switch the variables:

Solve for y:

Convert y term to a fraction to see how to simplify more easily:

$x+5=\frac{5}{2}y$

$\mathbf{y=\frac{2(x+5)}{5}}$

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Question

Find the inverse function of $6x-\frac{2}{3}y=4$ .

Answer

$6x-\frac{2}{3}y=4$

First convert to slope intercept form:

To find the inverse we switch the variables and solve for y again:

$\mathbf{y=\frac{x+6}{9}}$

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Question

Find the inverse function and simplify your solution:

Answer

To find the Inverse function of:

1. Replace with :

2. Switch the variables and :

3. Solve for :

$y = \frac{x - 21}{3}$

4. Simplify:

$y = \frac{1}{3}x - 7$

5. Replace with $f^{-1}(x)$ and the final solution is:

$f^{-1}(x) = \frac{1}{3}x - 7$

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