Composition of Functions

In mathematics, function composition is an operation that takes two functions f and g and produces a function h such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x. That is, the functions f : X → Y and g : Y → Z are composed to yield a function that maps x in X to g(f(x)) in Z.

Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function is denoted g ∘ f : X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X. The notation g ∘ f is read as "g circle f ", "g round f ", "g about f ", "g composed with f ", "g after f ", "g following f ", "g of f", "f then g", or "g on f ", or "the composition of g and f ". Intuitively, composing functions is a chaining process in which the output of function f feeds the input of function g.

The composition of functions is a special case of the composition of relations, sometimes also denoted by $\circ$ .[1] As a result, all properties of composition of relations are true of composition of functions, though the composition of functions has some additional properties.

Composition of functions is different from multiplication of functions, and has quite different properties; in particular, composition of functions is not commutative.

Examples

Concrete example for the composition of two functions.

• Composition of functions on a finite set: If f = {(1, 1), (2, 3), (3, 1), (4, 2)}, and g = {(1, 2), (2, 3), (3, 1), (4, 2)}, then g ∘ f = {(1, 2), (2, 1), (3, 2), (4, 3)}, as shown in the figure.
• Composition of functions on an infinite set: If f: ℝ → ℝ (where ℝ is the set of all real numbers) is given by f(x) = 2x + 4 and g: ℝ → ℝ is given by g(x) = x³, then:

(f ∘ g)(x) = f(g(x)) = f(x³) = 2x³ + 4, and

(g ∘ f)(x) = g(f(x)) = g(2x + 4) = (2x + 4)³.

• If an airplane's altitude at time t is a(t), and the air pressure at altitude x is p(x), then (p ∘ a)(t) is the pressure around the plane at time t.

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 a 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 : real number defined by f(x) = 9 − x² and g : interval (mathematics)#Infinite endpoints defined by $g(x)={\sqrt {x}}$ can be defined on the interval [−3,+3].

Compositions of two real functions, the absolute value and a cubic function, in different orders, show a non-commutativity of composition.

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 $x\geq 0$. 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)⁻¹ = g⁻¹∘ f⁻¹.

Resources

• Intro to Composing Functions, Khan Academy
• Compositions of Functions, Lumen Learning
• Composition of Functions, Math Is Fun

Licensing

Content obtained and/or adapted from:

• Function composition, Wikipedia under a CC BY-SA license