funktionkomposition

Function composition, in mathematics, is the operation of forming a new function by applying one function to the result of another. If f is a function from B to C and g is a function from A to B, the composite function f ∘ g from A to C is defined by (f ∘ g)(x) = f(g(x)). The domain of f ∘ g consists of those x in A for which g(x) lies in the domain of f.

Notation and interpretation: The composite is typically read as “f after g” or “f composed with g”; the order matters, and in general f ∘ g ≠ g ∘ f. When multiple functions are involved, f ∘ g ∘ h denotes f ∘ (g ∘ h). There is a neutral identity function id_X that satisfies id_X(x) = x.

Properties: Composition is associative: f ∘ (g ∘ h) = (f ∘ g) ∘ h, where the domains align. The identity function acts as a unit for composition: f ∘ id_X = f and id_Y ∘ f = f.

Calculus: If f and g are differentiable, the chain rule gives (f ∘ g)'(x) = f'(g(x)) · g'(x).

Examples: Let g(x) = x + 1 and f(x) = x^2. Then (f ∘ g)(x) = (x + 1)^2. Another example: f(x) = sin x, g(x) = x^2 yields (f ∘ g)(x) = sin(x^2).

Applications: Function composition appears in analysis, dynamical systems, and functional programming, where complex operations are built by composing simpler ones. Repeated composition leads to iterates, denoted f^n.