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 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*.

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