Funkcia

Funkcia is the Slovak term for the mathematical concept of a function. In mathematics, a function is a relation that associates to every element of a set, called the domain, exactly one element of another set, called the codomain. Functions are often written as f: X → Y, with f(x) = y meaning that x in X maps to y in Y. The choice of domain and codomain matters; a function is defined only on its domain, and the rule of correspondence specifies the output for each input.

Examples include the function f(x) = x^2 from real numbers to real numbers, which has domain R and codomain R and satisfies f(3) = 9. The function g: R → R, g(x) = sin(x) maps real inputs to real outputs. A function can be described by its rule, a table, a graph, or a composition of other functions.

Types commonly discussed are injective (one-to-one), surjective (onto), and bijective (both). An injective function assigns distinct outputs to distinct inputs; a surjective function covers every element of the codomain; a bijective function provides a perfect pairing between domain and codomain. A partial function is not defined for every element of the input set.

In computing and logic, the idea of a function appears as a value-producing procedure or method, often with or without side effects. In mathematics, functions are central to calculus, algebra, and analysis, underpinning concepts such as limits, continuity, and differentiation.

Etymology and history: The term funkcia derives from Latin functio, via early modern European languages. The concept emerged in the 17th–18th centuries, with contributions from mathematicians such as Leibniz and Euler, and was later formalized within set theory by Cantor.