Function composition

In mathematics, the composition operator ${\displaystyle \circ }$ takes two functions, ${\displaystyle f}$ and ${\displaystyle g}$, and returns a new function ${\displaystyle f\circ g}$. When the composite function ${\displaystyle f\circ g}$ (pronounced "${\displaystyle f}$ of ${\displaystyle g}$") is evaluated at an input ${\displaystyle x}$, the result is ${\displaystyle (f\circ g)(x)=f(g(x))}$. That is, the function ${\displaystyle f}$ is applied after applying ${\displaystyle g}$ to ${\displaystyle x}$.

The composition of functions is a special case of the composition of relations, sometimes also denoted by ${\displaystyle \circ }$. As a result, all properties of composition of relations are true of composition of functions, such as associativity.