A composite function
, or a composition of functions
, combines two or more , which produces a new function. In a composition, the produced by one function are the of the other function. The composition of the functions
is denoted as
Performing the composition of two functions is similar to one function into the other. For example, let
must be substituted with
Substitute for and evaluate
makes sense only when the outputs of
belong to the of
Also, be aware that the composition of functions is not
commutative — that is, in general,
This can be checked with the same two functions.
Just as changing the order of the machines in a factory could alter the final product, changing the order in which the functions are applied could produce different outputs. For example, here
are different values.
Explore the parts of a composition by moving the magnifying glass.