fonction

A fonction is a foundational concept in mathematics and related disciplines. In its most general sense, a fonction is a rule that assigns to every element of a domain X exactly one element of a codomain Y. It is typically denoted as f: X → Y, with f(x) = y indicating that x in X maps to y in Y. The set X is called the domain, the target set Y the codomain, and the outputs {f(x) : x in X} form the fonction’s range.

Functions can be classified by properties such as injectivity (one-to-one), surjectivity (onto), and bijectivity (both). A fonction is invertible precisely when it is bijective, in which case an inverse function f^{-1}: Y → X exists. Function composition combines two fonctions to form a new one: (g ∘ f)(x) = g(f(x)).

Typical examples: f: R → R, f(x) = x^2 is a fonction whose domain is all real numbers but whose range is nonnegative reals, illustrating that the codomain and range can differ. Other examples include polynomial, exponential, and trigonometric functions, each with distinct properties and graphs.

Beyond pure mathematics, the concept appears in computing as a function (or method) that takes inputs (parameters) and returns a value. In many programming languages functions are first-class values, can be passed as arguments, and may carry state or be pure (without side effects).

Terminology: in some languages the term fonction is used for mathematical contexts in the French tradition; in English, the term function is standard. The notion underpins theories in analysis, algebra, probability, and computer science, and informs practical modeling across sciences.