Range of a function


In mathematics, the range of a function may refer to either of two closely related concepts:
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 range of may refer to the codomain or target set or to, the image of the domain of under . The image of a function is always a subset of the codomain of the function.
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, since is never negative if is real. For this function, if we use "range" to mean codomain, it refers to. When we use "range" to mean image, it refers to.
The image and the codomain can coincide. Consider the function, which inputs a real number and outputs its double. For this function, the codomain and the image are the same, so the word range is unambiguous; it is the set of all real numbers.