structural:scholar_edition
Properties

The composition of functions is always associative —a property inherited from the composition of relations. That is, if f, g, and h are composable, then f ∘ (g ∘ h) = (f ∘ g) ∘ h. Since the parentheses do not change the result, they are generally omitted.

In a strict sense, the composition g ∘ f is only meaningful if the codomain of f equals the domain of g; in a wider sense, it is sufficient that the former be an improper subset of the latter. Moreover, it is often convenient to tacitly restrict the domain of f, such that f produces only values in the domain of g. For example, the composition g ∘ f of the functions f : R → (−∞,+9] defined by f (x) = 9 − x and g :.

The functions g and f are said to commute with each other if g ∘ f = f ∘ g. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, | x | + 3 = | x + 3 | only when x ≥ 0. The picture shows another example.

The composition of one-to-one (injective) functions is always one-to-one. Similarly, the composition of onto (surjective) functions is always onto. It follows that the composition of two bijections is also a bijection. The inverse function of a composition (assumed invertible) has the property that (f ∘ g) = g ∘ f.

Derivatives of compositions involving differentiable functions can be found using the chain rule. Higher derivatives of such functions are given by Faà di Bruno's formula.

Composition of functions is sometimes described as a kind of multiplication on a function space, but has very different properties from pointwise multiplication of functions (e.g. composition is not commutative).