function

A function is a relation between two sets that assigns to every element of a domain exactly one element of a codomain. In mathematics, a function f from X to Y maps each x in X to a y in Y, written f(x) = y. The domain is the set of permissible inputs; the codomain is the target set, and the range is the set of actual outputs.

Functions can be classified by how they map inputs to outputs. An injective (one-to-one) function assigns distinct outputs to distinct inputs; a surjective (onto) function covers every element of the codomain; a bijective function is both injective and surjective and thus has an inverse function f⁻¹ that undoes the mapping. Functions can be composed: if g maps some domain to an intermediate set, then f ∘ g maps the original domain to the final codomain. The graph, formula, table, or rule may describe a function.

Examples include polynomial, exponential, logarithmic, and trigonometric functions, as well as constant and piecewise-defined functions. Piecewise functions use different formulas on different input ranges and can model a variety of real-world behaviors.

In computer science, a function (or method) is a named block of code that accepts inputs and returns a value. Functions may have side effects or be pure, depending on whether they modify state outside their scope. They can be first-class, meaning they can be passed as arguments, returned from other functions, or stored in data structures. Higher-order functions, closures, and recursion enable modular, abstract programming and expressive control structures.