functions

Functions are a fundamental concept in mathematics and computer science. In mathematics, a function f from a set X (the domain) to a set Y (the codomain) assigns to every element x in X exactly one element f(x) in Y. The notation f: X → Y and the value f(x) are standard. The domain specifies where the function is defined; the codomain is the target set. The range, or image, is the set of outputs actually produced, a subset of the codomain. A function is often described as a rule or a correspondence between inputs and outputs.

Key properties include injectivity (one-to-one), where different inputs yield different outputs; surjectivity (onto), where every element of the codomain has a preimage; and bijectivity, where both hold. A function is invertible on its domain when it is bijective, with an inverse function. Functions can be composed: if f: X → Y and g: Y → Z, then g ∘ f: X → Z. The graph of a real-valued function is the set of ordered pairs (x, f(x)); notions of monotonicity, continuity, and limits describe its behavior. Common examples include linear functions f(x) = ax + b, polynomial functions, exponential, logarithmic, and trigonometric functions.

In computer science, a function is a named block of code that accepts inputs and returns a value. Functions may be pure (no side effects) or impure; they can be higher-order, taking or returning other functions. Closures capture surrounding state. Functions underlie most programming languages and contrast with subroutines by emphasizing value return and composability.