# Algebra II : Inverse Functions

## Example Questions

### Example Question #1 : Inverse Functions

Which of the following represents ?

Explanation:

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 .

Here's where we switch:

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 :

This is our inverse function!

### Example Question #1 : Inverse Functions

What is the inverse of the following function?

Explanation:

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 :

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 :

is our inverse function,

### Example Question #3 : Inverse Functions

What is  ?

Explanation:

The question is essentially asking this: take  say that equals , then take , then whatever that equals, say , take . So, we start with ; we know that , so if we flip that around we know . 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 , though, and if we flip it around, we get , which is our answer.

### Example Question #4 : Inverse Functions

What is  ?

Explanation:

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 . 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 , so , and that is our answer.

### Example Question #5 : Inverse Functions

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

You do not flip a function over a line when finding its inverse.

Explanation:

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.

### Example Question #6 : Inverse Functions

Find the inverse of this function:

Explanation:

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.

turns into the following once the variables are switched:

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

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.

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

### Example Question #7 : Inverse Functions

Please find the inverse of the following function.

Explanation:

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

Becomes

Now we need to solve for :

Finally, we need to divide each side by 4.

This gives us our inverse function:

### Example Question #1 : Inverse Functions

Find the inverse of .

Explanation:

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

y must be isolated to finish the problem.

### Example Question #9 : Inverse Functions

Which one of the following functions represents the inverse of

A)

B)

C)

D)

E)

D)

E)

C)

A)

B)

C)

Explanation:

Given

Hence

Interchanging  with  we get:

Solving for  results in .

### Example Question #10 : Inverse Functions

What is the inverse of ?

Explanation:

Interchange the  and  variables and solve for .