Funktsiyasi

Funktsiyasi, or function, is a relation between two sets that assigns to every element in the domain exactly one element in the codomain. The first set is called the domain, and the second set is called the codomain. Functions can be described by formulas, graphs, tables, or verbal rules. The notation f: X → Y is typical, with f(x) denoting the image of x in Y. A function may be defined by its rule, its domain of definition, and its codomain.

A function is called total if every domain element has an output; otherwise it is a partial function. Functions can be classified by properties such as injective (one-to-one), surjective (onto), and bijective (both). An injective function maps distinct inputs to distinct outputs. A surjective function covers every element of the codomain as an image of some domain element. A bijective function is both injective and surjective, implying the existence of an inverse function.

Examples include arithmetic functions like f(x) = x^2 on the real numbers, which is not injective on the entire domain, and the sine function sine: ℝ → [−1, 1], which is surjective but not injective. In computer science, functions are fundamental building blocks for programs, enabling modularity, higher-order functions, and function composition. In higher mathematics, functions are generalized to mappings between sets, topological spaces, or other structures, with additional properties such as continuity, differentiability, or measurability.

Funktsiyasi as a concept is foundational across disciplines, providing a precise framework to model dependence and correspondence between quantities.