Range of a function

In mathematics, the range of a function may refer to either of two closely related concepts: In some cases the codomain and the image of a function are the same set; such a function is called surjective or onto.

The sets X and Y are called the domain and codomain of f, respectively.

The image of the function f is the subset of Y consisting of only those elements y of Y such that there is at least one x in X with f(x) = y.

As the term "range" can have different meanings, it is considered a good practice to define it the first time it is used in a textbook or article.

Older books, when they use the word "range", tend to use it to mean what is now called the codomain.

[1] More modern books, if they use the word "range" at all, generally use it to mean what is now called the image.

[2] To avoid any confusion, a number of modern books don't use the word "range" at all.

,[4] may refer to the codomain or target set

consisting of all actual outputs of

[5] As an example of the two different usages, consider the function

as it is used in real analysis (that is, as a function that inputs a real number and outputs its square).

In this case, its codomain is the set of real numbers

, but its image is the set of non-negative real numbers

For this function, if we use "range" to mean codomain, it refers to

; if we use "range" to mean image, it refers to

For some functions, the image and the codomain coincide; these functions are called surjective or onto.

which inputs a real number and outputs its double.

For this function, both the codomain and the image are the set of all real numbers, so the word range is unambiguous.

Even in cases where the image and codomain of a function are different, a new function can be uniquely defined with its codomain as the image of the original function.

is not surjective because only the even integers are part of the image.

the word range is unambiguous.