# Inverse Functions

Functions and are inverses if

for all and . If f has an inverse, it is often denoted . However, does not mean " "!

Example. and are inverses, since

if x is a real number.

Notice that but the inverse is not !

Example. Functions which are inverses "undo" one another. Thus, if f and are inverses and f takes 4 to 17, then must take 17 to 4.

In symbols,

Example. In some cases, it's possible to find the inverse of a function algebraically. I'll find the inverse of .

First, I'll write it as .

Next, switch x's and y's:

(This means you should replace each "x" with a "y" and replace each "y" with an "x".)

Now solve for y in terms of x:

Since I was able to solve for y in terms of x, the result is the inverse function: .

The procedure I used tells something about the relation between the graphs of a function and its inverse. Since the inverse is obtained by swapping x's and y's, the graph of is a mirror image of the graph of f across the line .

In the picture below, I've shown the graphs of , , and :

Example. Find the inverse of .

Let . Swap x's and y's to obtain

Solve for y:

Therefore, .

Example. Find the inverse of .

Let . Swap x's and y's to obtain

Solve for y:

Hence, .

Not every function has an inverse. For example, consider . Now , so should take 4 back to 2. But as well, so apparently should take 4 to -2. can't do both, so there is no inverse! The problem is that you can't undo the effect of the squaring function in a unique way.

On the other hand, if I restrict to , then it has an inverse function: .

A function f is one-to-one or injective if different inputs go to different outputs:

Example. is not one-to-one, because different inputs can produce the same output. For example,

On the other hand, is one-to-one. For suppose the inputs a and b produce the same output: . Then

Then

That is, the inputs a and b were the same to begin with.

A graph of a function represents a one-to-one function if every horizontal line hits the graph at most once.

A one-to-one function has an inverse: Since a given output could have only come from one input, you can undo the effect of the function.

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