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The term range is sometimes ambiguously used to refer to either the codomain or the image of a function. A codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph. [1] The set of all elements of the form f(x), where x ranges over the elements of the domain X, is ...
Similarly, the inverse image (or preimage) of a given subset of the codomain is the set of all elements of that map to a member of . The image of the function f {\displaystyle f} is the set of all output values it may produce, that is, the image of X {\displaystyle X} .
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. In some cases the codomain and the image of a function are the same set; such a function is called surjective or onto.
The image under f of an element x of the domain X is f(x). [6] If A is any subset of X, then the image of A under f, denoted f(A), is the subset of the codomain Y consisting of all images of elements of A, [6] that is, = {()}. The image of f is the image of the whole domain, that is, f(X). [17]
For a function :, the set Y is called the codomain: the set to which all outputs must belong. The set of specific outputs the function assigns to elements of X is called its range or image. The image of f is a subset of Y, shown as the yellow oval in the accompanying diagram.
The function is surjective, or onto, if each element of the codomain is mapped to by at least one element of the domain; that is, if the image and the codomain of the function are equal. A surjective function is a surjection. [1] Notationally:
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 other words, every element of the function's codomain is the image of at most one element of its domain. [2] The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.