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# Logarithms and Exponential Functions

# Rules for Inverse Functions

There are a few rules for whether a function can have an inverse, though. Even though *you* can by anything you want in life, a function doesn't have all the same freedoms in math. Maybe they should draft a constitution or something.

First of all, it's got to *be a ***function** in the first place. For a review of that, go here...or watch this video right here:

Second, that function has to be *one-to-one*. That is, for every *x*-value, there's got to be a unique *y*-value.

This is a one-to-one function.

This is not. Notice how multiple *x* values can yield the same *y* value.

Figuring out if a function is one-to-one is as simple as drawing a straight line. No, really—give it a shot. It's called the **horizontal line test**. Draw any function, and then draw a straight horizontal line through it. If there's anywhere that the line passes through the function more than once, it is not a one-to-one function.

This function passes the horizontal line test

### Sample Problem

Which of the following is not a one-to-one function? Try drawing them if you have trouble.

a. *x*^{2} + 4

b. -4*x*

c. 2^{x}

The answer is *a*. Because this function is *even*, or symmetric across the *y*-axis, the horizontal line test *fails*, and it is not one-to-one. Don't worry, the function won't be punished, it's just part of a different circle of friends.

If a function isn't one-to-one, though, there's a simple way to make it conform: remove the parts that fail the horizontal line test. For example, if we have the function *x*^{2} + 4 from the sample problem:

This function fails the horizontal line test. No re-takes either—bummer.

All we've got to do is restrict the *domain *of the function so that it does pass the test with flying colors.

Here we restrict the domain to either *x* < 0 or *x* > 0, and now it's one-to-one.