Funksjonen

Funksjonen in mathematics is a relation that assigns to every input from a domain exactly one output in a codomain. A function is often denoted f: X → Y, with the value at x written as f(x). The domain contains admissible inputs; the codomain is the set of possible outputs; the range is the subset of outputs produced by the function.

Common examples include f(x) = x^2 from the real numbers to the real numbers, and f: R → R, f(x) = e^x. Functions can be real-valued, complex-valued, or multivariable, like f(x, y) = x + y.

Key properties include injective (one-to-one), surjective (onto), and bijective (both). A bijection has an inverse function g such that g(f(x)) = x for all x in the domain, and f(g(y)) = y for all y in the codomain. Function composition, written (g ∘ f)(x) = g(f(x)), builds new functions from existing ones.

The graph of a function is the set of ordered pairs (x, f(x)) in the plane; for real-valued functions this yields a conventional curve. Topics such as limits, continuity, differentiability, and integrability describe the behavior of functions at and near points.

In computer science, a function is a block of code that takes inputs (arguments) and returns an output. Functions may be pure, with no side effects, or impure, and they may be named or anonymous (lambdas).

The concept dates to the 17th–18th centuries, with Leonhard Euler helping popularize the term. Today, functions are foundational in mathematics, science, engineering, and programming.