# Inverse Functions

## Introduction to Inverse Functions

Inverse functions are functions that undo another function. For example, the inverse function of y=x+1 is y=x-1. This is a very simple example but inverse
functions can be a lot more complicated. In this lesson, you will learn how to test for inverse functions, find inverse functions and how inverse functions
look on graphs.
## Testing for Inverse Functions

If you have recently look at the functions lesson here on Ultimate Maths, you may remember the vertical line test. This test allowed us to determine
whether or not an equation is a function. A similar test allows us to determine whether or not a function has an inverse function. This test is called the
horizontal line test. It is identical to the vertical line test, except that this time any horizontal line drawn through a graph should not cut it more than once.
So for each value of y, there can only be one value of x. Here are a few examples:
Even though some of these functions such as x² do not pass the horizontal line test, they can still have an inverse function for a restricted domain. For
example, x² has an inverse function for x>0 because if we were to omit the left part of the graph, it would pass the horizontal line test. The same also
applies to other functions such as sine and cosine.
## Finding Inverse Functions

The finding of inverse functions is quite easy if your algebra skills are good. The technique we use to find them is very simple. You just have to be able to
rearrange them. The steps are as follows. Example function: y=4x+2. The example is worked out on the right.
1.
Write out the function
2.
Make y=x and x=y.
3.
Rearrange the equation so that you get back to y=...
Another method is to perform step 3 first making x the subject
and then switch x and y. Both methods work.
## Inverse Functions on Graphs

Inverse functions have a unique properties on graphs. They
are the reflection of the pervious graph reflected in the line
y=x. SO whenever you are asked to sketch the inverse of a
function, just draw in the line y=x and reflect all points in it.
Here is an example for the inverse functions we have worked
out above.
## Conclusion

We hope you now master functions. Make sure to check back for more lessons soon. Also, have a look for more resources and different
lessons in our library.
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Chapter 20.2: Learning Outcomes
Students will be introduced to the concept of inverse functions!
Students will learn how to test for inverse functions!
Students will learn how to find inverse functions!
Students will learn about the graphical properties of inverse functions!

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