Composing Functions
To compose two functions
and
is to use the result of one function in the other. If
and
then to find to composeandwith
withbeing performed first, find
then find
We can write this as
or
In general, the composition ofwith
with being performed first is written
or
or simply
(this is distinct from the product ofandwritten
or
).
You may think ofbeing used as the argument of
so that if
then
then if
The composition of functions has some important properties.
If a function
is the inverse of a functionso that
then
and
Composition of functions is associative so that
If
and
then
and 
We can compose functions graphically.
and
as shown below,
Then
and
So that