Range of a function

In mathematics, the range of a function may refer to either of two closely related concepts:

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.

TerminologyEdit

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][2] More modern books, if they use the word "range" at all, generally use it to mean what is now called the image.[3] To avoid any confusion, a number of modern books don't use the word "range" at all.[4]

Elaboration and exampleEdit

Given a function

 

with domain  , the range of  , sometimes denoted   or  ,[5][6] may refer to the codomain or target set   (i.e., the set into which all of the output of   is constrained to fall), or to  , the image of the domain of   under   (i.e., the subset of   consisting of all actual outputs of  ). The image of a function is always a subset of the codomain of the function.[7]

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

In many cases, the image and the codomain can coincide. For example, consider the function  , which inputs a real number and outputs its double. For this function, the codomain and the image are the same (both being the set of real numbers), so the word range is unambiguous.

See alsoEdit

Notes and ReferencesEdit

  1. ^ Hungerford 1974, page 3.
  2. ^ Childs 1990, page 140.
  3. ^ Dummit and Foote 2004, page 2.
  4. ^ Rudin 1991, page 99.
  5. ^ "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-28.
  6. ^ Weisstein, Eric W. "Range". mathworld.wolfram.com. Retrieved 2020-08-28.
  7. ^ Nykamp, Duane. "Range definition". Math Insight. Retrieved August 28, 2020.

BibliographyEdit