### All Algebra II Resources

## Example Questions

### Example Question #1 : Inverse Functions

Which of the following represents ?

**Possible Answers:**

**Correct 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 .

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 #2 : Inverse Functions

What is the inverse of the following function?

**Possible Answers:**

**Correct 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 :

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 ?

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct 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 . 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?

**Possible Answers:**

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

**Correct 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.

### Example Question #6 : Inverse Functions

Find the inverse of this function:

**Possible Answers:**

**Correct 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.

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.

**Possible Answers:**

**Correct answer:**

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 #8 : Inverse Functions

Find the inverse of .

**Possible Answers:**

**Correct answer:**

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)

**Possible Answers:**

D)

A)

C)

B)

E)

**Correct answer:**

C)

Given

Hence

Interchanging with we get:

Solving for results in .

### Example Question #10 : Inverse Functions

What is the inverse of ?

**Possible Answers:**

**Correct answer:**

Interchange the and variables and solve for .

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