# Composition and inverses

A function takes one number and gives you another.
You can then take this output and use it as the input to another function,
to get yet another number.
This is **composition** of functions.
## Definition

If *f* and *g* are functions,
then their **composite** is also a function,
denoted *f* ∘ *g*,
which I usually read as ‘*f* after *g*’.
It may be defined as follows:
Notice that you do *g* first and *f* afterwards.
The two composites
*f* ∘ *g* and *g* ∘ *f*
are not usually the same.
Sometimes it helps to use different variables for the different functions.
That is:

- Let
*y* be *g*(*x*);
- let
*z* be *f*(*y*);
- then
*z* = *f*(*g*(*x*)) =
(*f* ∘ *g*)(*x*).

## Domains of composites

If *f* and *g* are both defined for every real number,
then so is *f* ∘ *g*.
But in general, the domain of *f* ∘ *g*
is part of the domain of *g*.
We have: - dom (
*f* ∘ *g*) =
{*x* |
*x* ∈ dom *g*,
*g*(*x*) ∈ dom *f*}; that is,
*x* belongs to the domain of *f* ∘ *g*
if and only if *x* belongs to the domain of *g*
and *g*(*x*) belongs to the domain of *f*.

If you simplify the expression for *f* ∘ *g*,
then this will only help with the second condition;
be sure to go back and check the domain of *g*!
## Inverse functions

Two functions *f* and *g*
are **inverse functions** of each other if:
*f*(*g*(*x*)) = *x*
whenever *x* ∈ dom *g*, and
*g*(*f*(*x*)) = *x*
whenever *x* ∈ dom *f*.

Not every function has an inverse function!
If *f* does have an inverse function, then it only has one;
we call *f* **one-to-one**
and denote its unique inverse function by *f*^{−1}.
**Warning:**
*f*^{−1} does *not* mean 1/*f*!

There are two ways to caclulate *f*^{−1}:

- Start with
*x* = *f*(*y*) and solve for *y*;
this gives you *y* = *f*^{−1}(*x*).
- Start with
*y* = *f*(*x*) and solve for *x*;
this gives you *x* = *f*^{−1}(*y*).

If you don't get a single solution when you solve the equation,
then *f* is *not* one-to-one, and it has no inverse.
We have:

*f*^{−1}(*f*(*x*)) = *x*
whenever *x* ∈ dom *f*, and
*f*(*f*^{−1}(*y*)) = *y*
whenever *y* ∈ ran *f*.

## Graphs of inverse functions

The graphs of inverse functions are related by switching *x* and *y*.
In particular:
- The range of
*f*
is the same as the domain of *f*^{−1};
- the range of
*f*^{−1}
is the same as the domain of *f*.

A function is one-to-one
if and only if its graph satisfies the Horizontal Line Test:
every horizontal line goes through the graph at most once.

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This web page was written in 2011 and 2012 by Toby Bartels,
last edited on 2012 April 16.
Toby reserves no legal rights to it.
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