Funktioners

Funktioner (the Danish term for functions) are a central concept in mathematics and computer science, describing a relationship that assigns outputs to inputs. In mathematics, a function f maps a domain X to a codomain Y, written as f: X → Y, and it assigns to every element x in X exactly one element f(x) in Y. The domain is the set of allowed inputs, and the range or image is the set of possible outputs produced by f(X). A function is total if every x in X has an output; if some x produce no output, the function is partial.

Key properties describe how inputs relate to outputs. An injective (one-to-one) function assigns distinct outputs to distinct inputs. A surjective (onto) function covers its codomain, so every element of Y is the image of some x in X. A bijective function is both injective and surjective, and it has an inverse f^{-1}: Y → X that undoes the original mapping.

Common operations on funktioner include composition and restriction. If f: X → Y and g: Y → Z, the composite g ∘ f: X → Z maps x to g(f(x)). The identity function id_X: X → X maps every element to itself. Inverse functions exist precisely for bijections, providing a way to recover inputs from outputs.

In computing, funktioner are implemented as blocks of code that take inputs and return outputs. They can be pure (no side effects) or impure, and many languages treat function values as first-class, allowing higher-order usage such as passing functions as arguments or returning them from other functions.

Funktioner also appear in applied fields such as statistics, economics, and engineering to model relationships, describe dependencies, and analyze systems based on input–output mappings.