Wikipedia / Range of a function
Wikipedia / 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 This new function is surjective.
Definitions
Given two setss and , a binary relation between and is a function (from to ) if for every element in there is exactly one in such that relates to . The sets and are called the domain and codomain of , respectively. The image of the function is the subset of consisting of only those elements of such that there is at least one in with .
Usage
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. More modern books, if they use the word "range" at all, generally use it to mean what is now called the image. To avoid any confusion, a number of modern books don't use the word "range" at all.
Elaboration and example
Given a function
with domain,). The image of a function is always a subset of the codomain of the function.
as it is used in real analysis (that is, as a function that inputs a real number is real. For this function, if we use "range" to mean codomain; if we use "range" to mean image.
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. For example, as a function from the integers is not surjective because only the even integers whose domain is the integers and whose codomain is the even integers is the word range is unambiguous.
See also
Notes and references
Bibliography