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A function f from X to Y. The blue oval Y is the codomain of f. The yellow oval inside Y is the image of f, and the red oval X is the domain of f. In mathematics, a codomain or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X → Y.
A function f from a set X to a set Y is an assignment of one element of Y to each element of X. The set X is called the domain of the function and the set Y is called the codomain of the function. If the element y in Y is assigned to x in X by the function f, one says that f maps x to y, and this is commonly written = ().
is a function from domain X to codomain Y. The yellow oval inside Y is the image of . Sometimes "range" refers to the image and sometimes to the codomain. In mathematics, the range of a function may refer to either of two closely related concepts: the codomain of the function, or; the image of the function.
For every function f, let X be a subset of the domain and Y a subset of the codomain. One has always X ⊆ f −1 (f(X)) and f(f −1 (Y)) ⊆ Y, where f(X) is the image of X and f −1 (Y) is the preimage of Y under f. If f is injective, then X = f −1 (f(X)), and if f is surjective, then f(f −1 (Y)) = Y. For every function h : X → Y, one ...
In mathematics, a surjective function (also known as surjection, or onto function / ˈ ɒ n. t uː /) is a function f such that, for every element y of the function's codomain, there exists at least one element x in the function's domain such that f(x) = y. In other words, for a function f : X → Y, the codomain Y is the image of the function ...
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication.
In mathematics, an injective function (also known as injection, or one-to-one function [1]) is a function f that maps distinct elements of its domain to distinct elements of its codomain; that is, x 1 ≠ x 2 implies f(x 1) ≠ f(x 2) (equivalently by contraposition, f(x 1) = f(x 2) implies x 1 = x 2).
A complex-valued function of a real variable may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing complex values. If f(x) is such a complex valued function, it may be decomposed as f(x) = g(x) + ih(x), where g and h are real-valued functions. In other words ...