Function composition

Definition

Let $f \colon A \to B$ and $g \colon B \to C$ be functions. The composition of $g$ and $f$ is the function $g \circ f \colon A \to C$ with $$(g \circ f)(a) = g(f(a))$$ for each $a \in A$.

Remarks

Read $g \circ f$ as "$g$ after $f$": $f$ acts first. In general $g \circ f \neq f \circ g$. For $f(x) = x + 1$ and $g(x) = 2x$ on $\mathbb{R}$, $(g \circ f)(x) = 2x + 2$ and $(f \circ g)(x) = 2x + 1$. Composition is associative. See associativity of composition.

Explanations

L1 human

Composition chains two machines: the output of $f$ goes into $g$. For $g \circ f$ to make sense, the values of $f$ must lie in the domain of $g$.