# Precalculus : Inverse Functions

## Example Questions

### Example Question #14 : Find The Inverse Of A Function

Find the inverse of .

Explanation:

To find the inverse of the function, we simply need to switch the values of  and  and solve for .

Switching  and , we can write the function as:

We now subtract to solve for :

### Example Question #15 : Find The Inverse Of A Function

Find the inverse of .

Explanation:

To find the inverse of this function, we switch  and  in the function:

We now solve for :

### Example Question #11 : Find The Inverse Of A Function

Find the inverse of .

Explanation:

To find the inverse of this function we can switch the  and  variables and solve for .

First, switch  and  in the function:

Now, solve for :

### Example Question #17 : Find The Inverse Of A Function

Find the inverse of .

Explanation:

To find the inverse of the function, we switch  and  in the function.

We can now find our answer by solving for :

### Example Question #18 : Find The Inverse Of A Function

Find the inverse of .

Explanation:

To find the inverse of the function, we swtich  and  in the function.

Solve for :

### Example Question #19 : Find The Inverse Of A Function

Find the inverse of this function:

Explanation:

In order to have the inverse of a function, the new function must perform the inverse opperations in the opposite order. One way to ensure that is true is to consider the case of , switch x and y, then solve for y.

in this case becomes .

Our first step in solving is to take the reciprocal power on each side.

The reciprocal of 5 is , so we'll take both sides to the power of 0.2:

Now divide by 2:

Note that the answer has the correct inverse opperations, it is just in the wrong order - first you divide by 2, then you take x to the power of 0.2.

### Example Question #20 : Find The Inverse Of A Function

Find the inverse of this function:

Explanation:

In order to have the inverse of a function, the new function must perform the inverse opperations in the opposite order. One way to ensure that is true is to consider the case of , switch x and y, then solve for y.

In this case, becomes  and we solve for y.

subtract 1 from both sides

square both sides

now we will take both sides to the power of -1, in other words flip each side to the reciprocal.

We can consider to be

### Example Question #21 : Inverse Functions

Find the inverse of the follow function:

Explanation:

To find the inverse, substitute all x's for y's and all y's for x's and then solve for y.

### Example Question #22 : Inverse Functions

Find the inverse function.

Find the inverse function  of the function

None of these answers are correct.

Explanation:

To find an inverse function, switch x and y variables and solve again for y. The new function is the inverse. f(x) can be called y. To check your answer, you can insert either function into the x variable of the other, and the equations should both solve to equal x.

### Example Question #23 : Inverse Functions

Find the inverse of this function:

Explanation:

To find the inverse of a function like this, switch the x and y variables (thereby "inverting" the function) and then solve for x.

First we would set up:

And then let's factor:

And take the square root of both sides:

Leaving us with the final answer: