Range of a function
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 exampleEdit
Given a function
with domain , the range of , sometimes denoted or , 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.
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.
Notes and ReferencesEdit
- Hungerford 1974, page 3.
- Childs 1990, page 140.
- Dummit and Foote 2004, page 2.
- Rudin 1991, page 99.
- "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-28.
- Weisstein, Eric W. "Range". mathworld.wolfram.com. Retrieved 2020-08-28.
- Nykamp, Duane. "Range definition". Math Insight. Retrieved August 28, 2020.
- Childs (2009). A Concrete Introduction to Higher Algebra. Undergraduate Texts in Mathematics (3rd ed.). Springer. ISBN 978-0-387-74527-5. OCLC 173498962.
- Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley. ISBN 978-0-471-43334-7. OCLC 52559229.
- Hungerford, Thomas W. (1974). Algebra. Graduate Texts in Mathematics. 73. Springer. ISBN 0-387-90518-9. OCLC 703268.
- Rudin, Walter (1991). Functional Analysis (2nd ed.). McGraw Hill. ISBN 0-07-054236-8.