Composite Functions and Inverse Functions

Composite function. 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. Composition is used constantly in business modelling: for example, if $q$ is a function giving the quantity produced as a function of labour hours, and $C$ is a function giving total cost as a function of quantity produced, then $C \circ q$ gives total cost directly as a function of labour hours.

Composition is generally not commutative — that is, $f \circ g$ and $g \circ f$ are, in general, two entirely different functions, even when both happen to be defined. It is always necessary to compute both separately rather than assuming they agree.

Inverse function. If $f: A \to B$ is bijective (Section 4), then for every $b \in B$ there is exactly one $a \in A$ with $f(a)=b$. This lets us define a new function, the inverse function $f^{-1}: B \to A$, by $f^{-1}(b) = a$ whenever $f(a)=b$. In effect, $f^{-1}$ simply reverses every ordered pair of $f$: if $(a,b) \in f$, then $(b,a) \in f^{-1}$.

An inverse function undoes what the original function does, in the precise sense that

$f\big(f^{-1}(y)\big) = y \quad \text{for every } y \text{ in the domain of } f^{-1}$ …