**Video Transcript**

##
Transcript
**Inverse Functions**

A not-so-long time ago, in a galaxy not too far away, two intrepid heroes are traveling across the universe. Hans Olus and his crew have received instructions on how to reach the inverse universe. Their instructions are in the form of a function; but, to get to the inverse universe, they need to find the **inverse** of the **function**. The crew can only initiate their ship's warp drive if the function and its inverse meet, so accuracy is paramount.

### Inverse Functions

Some of the crew members do not know what an **inverse function** is, let alone how to find it. If you remember, for a **function f(x)**, every **‘x’ value**, produces one and only one **‘y’ value**. Simply speaking, the inverse of a function is a way to “undo” a function. In other words, if you are given a 'y' value, an inverse function will give you back your original 'x' value. For every point, finding the inverse is easy. You just switch the positions of the ‘x’ and ‘y’ coordinates. So, for example, if we’re given the point 'P', (5, 3), the inverse of this point is P-1(3, 5). That was simple, but what about a function? Finding a function's inverse is just a tad bit trickier.

### Vertical and Horizontal Line Test

First, you should know about the **vertical line test**. An **equation** is a function if it passes the vertical line test. Using a line **parallel** to the **y-axis**, if the line **crosses** the **graph** of your **target equation** at one and only one point for each 'x' value, then you can be sure the equation is a function. If an equation does not pass the vertical line test, then it is NOT a function.

Let's see if this equation passes the vertical line test. All **linear equations**, or equations with **degree 1**, pass the vertical line test. This linear equation is no different. It passes with flying colors. Now we can write this equation in **function notation**. In order to find out if the **inverse** of an **equation** also is a function, it has to pass the **horizontal line test**. Let's check this linear function to see if it'll pass. Yup! Looks good! So now we know that this function has an inverse function, but how do we find it?

### Finding the Inverse Function

We can do this algebraically by following some simple steps. Hans Olus and his crew have to find the **inverse function** of f(x) = 3x - 6. First, we'll put it into 'y =' notation. Next, we switch every 'x' in our original function with a 'y' and every 'y' with an 'x'. What happens next? You guessed it! We have to **isolate 'y'**, just like we've done countless times before.

And there you have it, ladies and gentlemen! Our first inverse equation! There is an interesting fact about functions and their inverses. If y = x, you get the following graph. You will notice that the function and its inverse **reflect** about this line. Inverse functions are written using the following notation. This is read, **"the inverse of F of X"**.
For our original equation, if we plug in values like 0, 1, 2, 3, 4 and so on we get the following values for 'y': If we plug these values into the inverse function we found, we should get back our original numbers. Let's do that now. The 'y' values for our original equation are -6, -3, 0, 3, and 6. Substituting these values into our inverse function we get...0, 1, 2, 3, 4! As you can see, the 'x' and 'y' values of the original and the inverse function are just interchanged.

### Summary

To find the **inverse function f-1(x)** of a function f(x) you first have to **replace** f(x) with 'y' in the equation for f(x). Then you have to change all 'y's to 'x's and all 'x's to 'y's. Now you're ready to **solve the equation** for 'y'. Finally, replace the 'y=' notation with the inverse function notation, f⁻¹(x).

The crew pass their findings back to Hans Olus and he gets the warp drive ready to go to the parallel universe, Inversium. On no! What happened?! Seems like not only functions can have inverses in Inversium...

## 1 comment

Very cool!