Composite Functions

Given two functions $f : A \to B$ and $g : B \to C$, the composite function $g \circ f : A \to C$ (read "$g$ composed with $f$", or "$g$ of $f$") is defined by first applying $f$ and then applying $g$ to the result:

For this composition to make sense, the codomain of $f$ must match (or be contained in) the domain of $g$ — the output of the first function must be a valid input to the second.

The single most important rule of composition is that it is generally not commutative: $f \circ g$ and $g \circ f$ are, in general, two entirely different functions, even when both are defined. They must always be worked out separately, never assumed equal. For example, with $f(x)=x+1$ and $g(x)=x^2$,

which are clearly different.

Composition is, however, associative: $(h\circ g)\circ f = h\circ(g\circ f)$, so a chain of three or more functions may be grouped in any way. Composing any function $f$ with the identity function $I(x)=x$ leaves it unchanged: $f\circ I = I\circ f = f$. …