funktsioone

Funktsioone (Estonian for function) is a fundamental concept in mathematics and computer science. In its formal sense, a funktsioone is a relation that assigns to each element of a domain exactly one element of a codomain. The domain is the set of inputs; the codomain is the set of possible outputs, while the range is the subset of outputs actually produced by the function.

Functions are typically denoted by letters such as f, g, or h, and written as f: X -> Y, with f(x) = y. Here x must belong to X, and the resulting y must belong to Y. Functions can be described by formulas, tables, or algorithms, and they can be visualized by graphs consisting of ordered pairs (x, f(x)).

Functions are classified by several properties. An injective (one-to-one) function assigns distinct outputs to distinct inputs; a surjective (onto) function reaches every element of the codomain; a bijective function is both injective and surjective and thus has an inverse. On ordered domains, one may discuss monotonicity; on topological spaces, continuity is another important property. The composition of functions, if f: X -> Y and g: Y -> Z, yields a new function g ∘ f: X -> Z defined by (g ∘ f)(x) = g(f(x)). A function is invertible precisely when it is bijective, with an inverse f^{-1}: Y -> X.

In computer science, a funktsioone is a named block of code that maps inputs to outputs. Functions can be pure, producing no side effects, or impure, potentially altering state. Many programming languages support higher-order functions that take other functions as arguments or return them as results.

Funktsioone are central across disciplines, underpinning mathematical modeling, analysis, and computation, and providing a reusable and expressive tool for describing relationships and processes.