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 for all x,

Notice that the inverse of 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. Let .

First, switch x's and y's:

Solve for y in terms of x. The result is :

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 :

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:

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.

Calculus provides an easy way of telling when a function is one-to-one, and hence when a function has an inverse.

\boxedtext{2}{A function which is *increasing* on an interval
is one-to-one, (and therefore has an inverse). A function which is
{\it decreasing} on an interval is one-to-one (and therefore has an
inverse).}

A differentiable function increases on an interval if its derivative is positive, and decreases on an interval if its derivative is negative.

* Example.* Let . Then

for and for . So f increases for and decreases for .

It follows that f is one-to-one (and has an inverse) on or on .

As you can see, either the left half of the graph or the right half of the graph would pass the horizontal line test. But the whole graph does not.

You can use implict differentiation to find the derivative of the inverse of a function. Let . This means , so differentiating implicitly,

That is,

* Example.* The * inverse
sine* function satisfies

The derivative of is

Let . Then :

Thus, , so

* Example.* If and , then

* Example.* Suppose , so .
Differentiating directly,

To use the formula for the derivative of the inverse, note that . Therefore,

The results are the same.

* Example.* Suppose that and .
Find .

Since , . So

* Example.* Let . Notice that . Find .

First, . Then

Copyright 2009 by Bruce Ikenaga