Editing Range of a function (section)

==Elaboration and example==
Given a function
:<math>f \colon X \to Y</math>
with [[domain of a function|domain]] <math>X</math>, the range of <math>f</math>, sometimes denoted <math>\operatorname{ran}(f)</math> or <math>\operatorname{Range}(f)</math>,<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Range|url=https://mathworld.wolfram.com/Range.html|access-date=2020-08-28|website=mathworld.wolfram.com|language=en}}</ref> may refer to the codomain or target set <math>Y</math> (i.e., the set into which all of the output of <math>f</math> is constrained to fall), or to <math>f(X)</math>, the image of the domain of <math>f</math> under <math>f</math> (i.e., the subset of <math>Y</math> consisting of all actual outputs of <math>f</math>). The image of a function is always a subset of the codomain of the function.<ref>{{Cite web|last=Nykamp|first=Duane|date=|title=Range definition|url=https://mathinsight.org/definition/range|archive-url=|archive-date=|access-date=August 28, 2020|website=Math Insight}}</ref>

As an example of the two different usages, consider the function <math>f(x) = x^2</math> 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 <math>\mathbb{R}</math>, but its image is the set of non-negative real numbers <math>\mathbb{R}^+</math>, since <math>x^2</math> is never negative if <math>x</math> is real.  For this function, if we use "range" to mean ''codomain'', it refers to <math>\mathbb{{\displaystyle \mathbb {R} ^{}}}</math>; if we use "range" to mean ''image'', it refers to <math>\mathbb{R}^+</math>.

For some functions, the image and the codomain coincide; these functions are called ''[[surjective function|surjective]]'' or ''onto''. For example, consider the function <math>f(x) = 2x,</math> 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 [[integer]]s to the integers, the doubling function <math>f(n) = 2n</math> is not surjective because only the [[even integers]] are part of the image. However, a new function <math>\tilde{f}(n) = 2n</math> whose domain is the integers and whose codomain is the even integers ''is'' surjective. For <math>\tilde{f},</math> the word ''range'' is unambiguous.