funksionet

Funksionet, or functions, are a foundational concept in mathematics and computer science. Formally, 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 pair (x, f(x)) traces the relation, and the image f(X) is the range—a subset of Y. Notation f: X → Y denotes the domain and codomain, though many discussions treat the codomain loosely as the target of the mapping.

Functions can be described by a rule, a table of values, or a graph. Examples include f(x) = x^2 with domain R; if the domain is restricted to x ≥ 0, f is injective and has an inverse f^{-1}(y) = sqrt(y) on its image. Inverse functions exist precisely when the function is bijective (both injective and surjective).

Common types include linear functions f(x) = ax + b; polynomial, rational, exponential, logarithmic, and trigonometric functions. Key properties include injectivity (one-to-one), surjectivity (onto), and bijectivity (both). The graph of a function is the set of points (x, f(x)) in the Cartesian plane; domain and range describe the allowable inputs and outputs, while the codomain specifies the intended target.

In calculus, continuity, differentiability, and integrability describe how functions behave. Composition of functions, (g ∘ f)(x) = g(f(x)), yields new functions, and inverse functions satisfy f^{-1}(f(x)) = x.

In computer science, a function is a named block of code that maps inputs to outputs; in functional programming, functions can be treated as values and composed without side effects.

Applications span mathematics, physics, economics, statistics, and data processing, where funksionet model relationships, transform data, and guide algorithms.