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:

- (
*f*∘*g*)(*x*) =*f*(*g*(*x*)).

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*).

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 only 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*.

Go back to the course homepage.

This web page was written by Toby Bartels, last edited on 2021 August 30. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-1300/2021FA/composition/`

.