Composition of functions

Let $f\colon A\to B$ and $g\colon B\to C$ be functions with matching codomain /$\,$domain . Their composition is the function

Composition is associative, and the identity function acts as a two-sided identity for composition.

Examples:

• If $f(x)=x^2$ and $g(x)=x+1$ on $\mathbb{R}$, then $(g\circ f)(x)=x^2+1$ while $(f\circ g)(x)=(x+1)^2$.
• If $\iota\colon A\to B$ is an inclusion and $h\colon B\to C$, then $h\circ\iota$ is the restriction of $h$ to $A$.