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{{Other uses|Range (disambiguation){{!}}Range}}


[[Image:Codomain2.SVG|right|thumb|250px|<math>f</math> is a function from [[domain of a function|domain]] '''X''' to [[codomain]] '''Y'''. The smaller oval inside '''Y''' is the [[image (mathematics)|image]] of <math>f</math>.  Sometimes "range" refers to the codomain and sometimes to the image.]]


In [[mathematics]], the '''range''' of a [[Function (mathematics)|function]] refers to either the '''[[codomain]]''' or the '''[[image (mathematics)|image]]''' of the function, depending upon usage. The codomain is a set containing the function's outputs, whereas the image is the part of the codomain which consists only of the function's outputs.  
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For example, the function <math>f(x) = x^2</math> is often described as a function from the [[real number]]s to the real numbers, meaning that its codomain is the set of real numbers '''R''', but its image is the set of non-negative real numbers, as <math>x^2</math> is never negative if <math>x</math> is real. Some books use the term range to indicate the codomain '''R'''. These books call the actual output of the function the image. This is the current usage for [[Range (computer science)|range in computer science]]. Other books use the term range to indicate the image, that is the non-negative real numbers. In this case, the larger set containing the range is called the codomain.<ref>Walter Rudin, ''Functional Analysis'', Second edition, p. 99, McGraw Hill, 1991, ISBN 0-07-054236-8</ref> This usage is more common in modern mathematics.
 
==Examples==
Let ''f'' be a function on the [[real numbers]] <math>f\colon \mathbb{R}\rightarrow\mathbb{R}</math> defined by <math>f(x) = 2x</math>. This function takes as input any real number and outputs a real number two times the input. In this case, the codomain and the image are the same (''i.e.'', the function is a [[Surjective function|surjection]]), so the range is unambiguous; it is the set of all real numbers.
 
In contrast, consider the function <math>f\colon \mathbb{R}\rightarrow\mathbb{R}</math> defined by <math>f(x) = \sin(x)</math>. If the word "range" is used in the first sense given above, we would say the range of ''f'' is the codomain, all real numbers; but since the output of the [[sine]] function is always between -1 and 1, "range" in the second sense would say the range is the image, the closed interval from -1 to 1.
 
==Formal definition==
 
Standard mathematical notation allows a formal definition of range.
 
In the first sense, the range of a function must be specified; it is often assumed to be the set of all real numbers, and {''y'' | there exists an ''x'' in the domain of ''f'' such that ''y'' = ''f''(''x'')} is called the image of ''f''.
 
In the second sense, the range of a function ''f'' is {''y'' | there exists an ''x'' in the domain of ''f'' such that ''y'' = ''f''(''x'')}.  In this case, the codomain of ''f'' must be specified, but is often assumed to be the set of all real numbers.
 
In both cases, image ''f'' ⊆ range ''f'' ⊆ codomain ''f'', with at least one of the containments being equality.
 
==See also==
*[[Bijection, injection and surjection]]
*[[Rescaled range]]
*[[Range of a matrix]]
 
==References==
{{reflist}}
 
{{logic}}
 
{{DEFAULTSORT:Range (Mathematics)}}
[[Category:Functions and mappings]]
[[Category:Basic concepts in set theory]]
 
[[mzn:برد]]
[[de:Bild (Mathematik)]]

Latest revision as of 21:50, 10 October 2014


One theory is to give allowance for the comfort level of the racer himself or even to offer some freedom to him in choosing his o...

There has to be some explanations why when it comes to triathlon apparels, it is purely an individual preference of the speed. Well, naturally staff outfits mark the exemption. Depending on the agreement or the rulings as required by a staff, competitors are released or required with a single type of standard. In general, but competitors are allowed using their decision.

One idea is to give money for the level of comfort of the racer herself or even to provide some freedom to him in choosing his own apparel. Having the ability to decide for your own personal suit and items would do good to make the race more endurable, given that it is composed of three disciplined strength games.

And what more may be a lot better than having your custom triathlon clothing?

Yes, there may be good quality triathlon retailers both online and offline that may offer you their shirts, shorts and best wetsuits and accessories. However, there simply are not any standard sizes that would perfectly fit every racer. All of us have our unique anatomies. When looking for one and being so particular using the installation of the clothing would surely create some troubles.

Custom triathlon apparels might not give you anything that you"d desire in a suit but, it can provide all the things that you should have. An ideal match, resources that are of one"s personal selection and a design that you truly served conceptualized. Discover additional information on this partner web page - Navigate to this link: Archery Record | Black Berry Apps. It"s only therefore personal.

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