# Talk:Domain of a function

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## Removed expansion template

I've removed the {{expand}} template (and Oleg's comment that he had moved it from the main page). I think this article is, if anything, already too long for its subject matter; there's just not that much to say. --Trovatore 14:15, 6 January 2006 (UTC)

Good! :) Oleg Alexandrov (talk) 22:15, 6 January 2006 (UTC)

I would suggest adding a softer section at the beginning, or at least a disclaimer that this article assumes a fair comfort with set theory. I suspect many calculus and precalculus students (or their parents) that might want to look at an articles on domain and range would have a bit of trouble with this. I would try to write one but it looks like the consensus is not to expand the article. If it is worth it I would be glad to write something. Thenub314 13:13, 30 September 2006 (UTC)

If you can make the lead section more accessible (while keeping it accurate!) by all means do. That's not the sort of expansion we (if I may presume to speak for Oleg) are concerned about. "Expansion" suggests finding more to say, not saying it more understandably; I fear the only way to find more material about this topic is to let it devolve into trivia. --Trovatore 20:35, 30 September 2006 (UTC)

I find the opening far too difficult. 67.71.156.34 22:16, 5 October 2006 (UTC)

I conjecture that part of the confusion is from the fact that the article starts out with saying the codomain is the set of 'possible' output values and the range is the set of 'actual' output values. This is confusing to say the least, here 'possible' means all values that might be produced including those that cannot be produced for any input! This is only clear once the reader makes it all the way to 'actual outputs' in the next sentence. This would rather non-standard English usage of 'possible'. Wolfram Mathworld treats this topic point without creating confusion; see http://mathworld.wolfram.com/Codomain.html . An example or a link to one might help; there is a simple example at codomain. Perhaps some of the intro should be re-ordered? Many people probably make it through most of life, or at least secondary schooling, equipped with a fuzzy notion of range. Do we need to correct this just to explain the idea of domain? 24.226.31.7 05:47, 19 October 2006 (UTC)

## Confusing

I also found this article confusing. How is something "possible" if it's not "actual"? Thanks for the Wolfram link, but I must say I'm still a little confused. According to that definition, "A set within which the values of a function lie", the codomain of a function could be any number of sets provided that each set contains at least all of the function's actual/possible values. Is that correct? 4.252.2.193 03:35, 14 April 2007 (UTC)

That's right. Any set containing all the values is a valid codomain. You see, saying that "function's codomain is the real numbers" is the same as saying "function takes real values". Practically speaking it is much easier to say where a function takes values than what all the values are. Oleg Alexandrov (talk) 05:10, 14 April 2007 (UTC)

I've read this article as well, before comming to the talk and seeing your message, and I agree, it is very confusing for students. I've put the template up. -- penubag  08:42, 22 January 2008 (UTC)

Can you explain what you find confusing? If you don't, it's quite hard for anyone to try to remedy the problem. — Carl (CBM · talk) 12:31, 22 January 2008 (UTC)
It's confusing in a number of ways: It does not clearly explain exactly what is the domain. It does not clearly state the domain's relation with respect to the x-axis. When read from a student's POV, the article just sounds like mesh, no clear statements are generally being said, I would fix it, but I understand the difficulty of doing this. -- penubag  19:03, 23 January 2008 (UTC)
The domain of an abstract function will not be related to the x axis in any particular way. That would only apply to functions defined on the real line. The article begins with the sentence "In mathematics, a domain is most often defined as the set of values, D for which a function is defined." - which seems like a clear enough definition. I am also interested in improving the article; but I'm not sure exactly what you are seeing. — Carl (CBM · talk) 19:04, 23 January 2008 (UTC)
I agree with Carl here. The definition as given seems reasonably clear. Something might be added to the article stating that when considering the graphical representation of a function, particularly functions whose domain is some subset of the real numbers, the domain is often represented as a subset of the horizontal axis (often called the x-axis) of a Cartesian plane. Not sure that such an addition would be particularly helpful, though. Perhaps penubag can pick a particular sentence which is found to be confusing, and we can work from there. Paul August 21:26, 23 January 2008 (UTC
The definition provided is confusing for students. "In mathematics, a domain is most often defined as the set of values, D for which a function is defined." D? what's D? for a which a function is defined? Any function? Maybe a quick definition of function would be great. According to here there are 3 different definitions (but those provided are still rather unclear). The definition in Prentice Hall PreCalculus states that the domain of a function is the function's limit with respect to the x axis, while the range of the function is the graph's limit with respect to the y axis. Surely this should be somewhere stated in our article. Also this seems confusing (or that I find confusing), "A function that has a domain N is said to be a function over N, where N is an arbitrary set." What is N? We should state that it just a variable representing the domain. (is it?) It has a domain N to be a function over N? As in a fraction? A little confusing. Basically saying, only one that is really familiar with math could make since out of what's being said, as a student's POV, it is really hard to grasp this concept, perhaps fixing some terminology could alleviate it. -- penubag  04:20, 24 January 2008 (UTC)

I found some useful information here, maybe we could use some of it. -- penubag  04:23, 24 January 2008 (UTC)

The issue with the Mathworld refs is that they say the same thing as this article: "The term domain is most commonly used to describe the set of values D for which a function (map, transformation, etc.) is defined." "(1) The set of values for which a function is defined." I don't mind giving an example involving a function from the reals to the reals, but the concept of domain is for a general function, and in the end must be handled as such. — Carl (CBM · talk) 04:29, 24 January 2008 (UTC)
Yeah, I know, but we have the opportunity to fix ours to a concise level, I'm just not a math guy, so I wouldn't know how to best do this. -- penubag  04:56, 24 January 2008 (UTC)
A simple definition for the domain, that's all this article needs right now. WinterSpw (talk) 04:16, 16 September 2008 (UTC)

I've tried to implement some simplifications. Is the page, esp. intro, any better now? Zaslav (talk) 19:03, 15 March 2010 (UTC)

## new section?

I'm not going to add this just yet because I'm having a hard time expressing the idea here with clarity. I think we need to mention the terms "domain of definition" and "restricted domain" Here is an attempt:

### Domain of definition

Usually "domain" means "domain of definition", that is, the set of values for which the function is defined. However, in some contexts, such as complex analysis "domain" refers to a restricted domain. The restricted domain is a subset of the domain of definition. It can be chosen arbitrarily. Some texts use the phrase "domain of definition" for added clarity.

Thoughts? futurebird 01:21, 1 December 2007 (UTC)

"Domain of definition", while not widely used, is correct and is obviously a good term to introduce in the exposition. I am doing that. "Restricted domain" is more specialized and I think it's treated well in defining the restriction of a function, in the article. Zaslav (talk) 18:57, 15 March 2010 (UTC)

## Problematic image

I've commented out the image and caption that appear at the top of the page. It stated that the domain of f(x) = √x "is" [0,+∞), but this is only strictly true if we restrict the range of f(x) to not include complex numbers. I think this is very unhelpful, given that the subject of the article concerns such matters. 86.129.77.213 (talk) 17:39, 3 February 2009 (UTC)

## Requested move

The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the move request was moved to Domain of a function, which seems to have raised no real objections. Aervanath (talk) 15:56, 14 June 2009 (UTC)

Domain (mathematics)Domain (function) — make the title to the point: "(mathematics)" is ambiguous, since there are several mathematical meanings. -- Smjg (talk) 13:42, 5 June 2009 (UTC)

Originally requested as a {{db-move}}. Contested by "not non-controversial: (mathematics) allows for a broader subject base" [1]

Contrary to BF's comment, the other mathematical meanings of "domain" have their own pages, as indicated by both the beginning of Domain (mathematics) and the Domain disambiguation page. Because this page is about just one of these meanings, namely that which is to do with functions, it should be at Domain (function). The whole point of bracketed parts of article titles is to indicate which article is which, which the current title doesn't. -- Smjg (talk) 13:42, 5 June 2009 (UTC)

I agree that this topic isn't a sufficiently dominant use of "domain" (even in the context of mathematics) to have it here at domain (mathematics). Therefore domain (mathematics) should be a sub-disambiguation page or perhaps link back to the relevant subsection of domain.
I am not particularly happy with the name "domain (function)". It's not clear how to apply the criteria from WP:NCDAB in this case, but I think we could consider "domain of a function" to be "a more complete name (such as Delta rocket instead of Delta) that is equally clear and unambiguous". The problem with "(function)" is that a domain is not a function, and "function" is also not really a field of mathematics in which this definition is used.
Therefore I propose moving this article to domain of a function. --Hans Adler (talk) 14:17, 5 June 2009 (UTC)
If we're going to have it at something other than Domain (function), what do people think is best? Domain of a function, Function domain or something else? I'm not sure.... -- Smjg (talk) 14:52, 5 June 2009 (UTC)
Not Function domain, which would be a ring of functions (presumably a rather limited one, to avoid zero divisors). Domain (theory of functions)? Septentrionalis PMAnderson 15:02, 5 June 2009 (UTC)
Or maybe even Domain (functions). I'm not sure.... -- Smjg (talk) 19:23, 5 June 2009 (UTC)
No, I don't think function domain should even exist (a Google Scholar search brings up all sorts of unrelated things including "erectile function domain", and even after restricting to "Engineering, Computer Science, and Mathematics" I couldn't find a single relevant hit), and domain (functions) is not better than domain (function) since both have exactly the same problems. Function domain seems to have been the name of this article once. Then Michael Hardy renamed it to domain of a function, and then someone renamed it to its present name. (All this happened back in 2004.) I believe Michael Hardy was right. Is there something you don't like about domain of a function? --Hans Adler (talk) 20:27, 5 June 2009 (UTC)
OK, I don't think it's a big problem because morphisms are generalised functions, but this is equally an argument against "domain of a function".
Should we decide to keep the article at its current name, we can get rid of the third alternative meaning at the top by integrating it (in completely rewritten form) into this article. I don't think domain of discourse is a sufficiently important topic to have its own article. I would prefer it to be covered mostly in structure (mathematical logic), with a little summary here. It's really a generalisation of the domain of a function in that finite powers of the domain of discourse serve as domains for functions and relations in a structure. (It's also the codomain for the functions). --Hans Adler (talk) 22:14, 5 June 2009 (UTC)
It would be harmless at universe of discourse, where it belongs. Septentrionalis PMAnderson 18:51, 6 June 2009 (UTC)
This sounds as if it might make sense. I have never heard this term. "Domain of discourse", yes. "Universe", yes. "Universe of discourse", no. But Google Books gives me plenty of non-mathematical hits that seem to be related to the non-mathematical side of the domain of discourse article. --Hans Adler (talk) 23:32, 6 June 2009 (UTC)
Come to think of it, I know "universe of discourse" from a late-nineteenth century mathematician; but if it has survived in semantics, that's enough to have an article, no? Septentrionalis PMAnderson 20:43, 7 June 2009 (UTC)
It was used by Church and Tarski, but JSTOR suggests "universe of discourse" is current in the philosophy of mathematics. Septentrionalis PMAnderson 20:50, 7 June 2009 (UTC)
The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

## Confusing (bis)

The article starts with saying that for a function f : X → Y, X is the domain, and Y is the codomain. It then proceeds by saying that for a well-defined function every element of its domain must be carried to an element of its codomain. In other words, consistent with the practice of most mathematicians, the domain is the set of objects x for which f(x) is defined.

However, in the section of partial functions, it then states that:

"Most mathematicians, including recursion theorists, use the term "domain of f" for the set of all values x such that f(x) is defined. But some, particularly category theorists, consider the domain of a partial function f : X → Y to be X, irrespective of whether f(x) exists for every x in X."

Wait, doesn't this amount to the very same thing then? That is, how are the category theorists saying anything different when they say that the domain of f : X → Y is X given that the article says the very same thing at the very beginning?!?

So: if you want to stick to the practice of most mathematicians, I recommend starting the article either by: "the domain of a function f is the set of all objects x for which a function value f(x) is defined", or you say that "the domain of a function f : X → Y is the set of all elements x in X for which a function value f(x) is defined". Using the latter, you can then continue by saying: "for a well-defined (or total) function f : X → Y, the domain equals X, and for a partial function f : X → Y, the domain is a strict subset of X".

Similarly, instead of saying that the co-domain of a function f : X → Y is Y, you could define the codomain as the set of all elements y in Y for which there is some x in X for which f(x) = y, followed by saying that for a surjective (onto) function, the co-domain equals Y, but that for a non-surjective function, the co-domain is a strict subset of Y.

Wouldn't that be more consistent? — Preceding unsigned comment added by Bram28 (talkcontribs) 17:11, 28 November 2011 (UTC)

Partial functions are not used very often, and the terminology surrounding them is not completely standardised. If we want to be very formal, then the following are two common definitions of functions:
1. A function f from X to Y is a subset of X × Y such that for every x in X there is precisely one y in Y such that (x,y) is in f.
2. A function f from X to Y is a triple (X, F, Y), where F is a function from X to Y in the previous sense.
There are two corresponding definitions of partial functions:
1. A partial function f from X to Y is a subset of X × Y such that for every x in X there is at most one y in Y such that (x,y) is in f. Equivalently, it is a function from a subset of X to Y.
2. A partial function f from X to Y is a triple (X, F, Y), where F is a partial function from X to Y in the previous sense. Equivalently, it is a triple (X, F, Y), where F is a function from some subset of X to Y.
Your suggested definition for "codomain" is what is generally known as the image or range. These terms are totally non-ambiguous, so there is no incentive to introduce the same thing as a new, alternative meaning for the also non-ambiguous term codomain. Functions and partial functions in the first sense do not have codomains, the only have an image/range. Functions and partial functions in the second sense have both. The image/range (set of all y for which there is a corresponding x) is always a subset of the codomain Y.
The point of the second style of definition is precisely to codify what the domain and codomain of a function are, by making them explicitly part of the function as a mathematical object. For functions this is not really necessary, because X and the set of x for which f(x) is defined are one and the same. But for partial functions the two notions diverge, making the notion of the domain of a partial function ambiguous unless you know whether an author uses the first or second style of definition. Hans Adler 21:21, 28 November 2011 (UTC)

## Definition expansion

Could somebody expand the text

Any function can be restricted to a subset of its domain. The restriction of $g \colon A \to B$ to $S$, where $S \subseteq A$, is written $g|_S \colon S \to B$.

by merging in at least some of the content from Function (mathematics) section on Restrictions and Extensions and Restriction (mathematics)?

BCG999 (talk) 21:14, 3 December 2012 (UTC)

## Domain of a term?

Why are you guys only referring to functions here? Domains to my knowledge exist for mathematical terms (i. e. non-functions) as well! For instance, for $x \in \R$, the term $\sqrt{x-1}$ is only defined for $x >= 1$. Hence, its domain would be written like $\{x \in \R | x\geq 1\}$. The German Wikipedia article (see: de:Definitionsmenge) does have an extra section for the domain of a term; in WP:EN, this is obviously missing (or deemed irrelevant). -andy 77.190.37.140 (talk) 19:06, 16 March 2013 (UTC)

Apparently the German article defines a broader concept than the domain of a function. However, with the title "Domain of a function" it is not relevant to discuss expressions such as $\sqrt{x-1}$. Are there reliable sources in English that define the domain of a term or expression the way you are suggesting? Isheden (talk) 19:11, 17 March 2013 (UTC)
The domain of a term or expression is a nonsense if the term does not represent a function. If it represents a function, its domain is that of the function, and no further definition is needed. If the expression represents several functions, then there may be as many different domains. For the above example, if it is considered as a function from the reals into the reals, then the domain is what has been said. If functions from the reals or from the complexes into the complexes are considered, the domain is the whole real line or the whole complex plane. D.Lazard (talk) 09:49, 23 April 2013 (UTC)
Possibly the IP user is referring to what is called natural domain in this article: "The natural domain of a formula is the set of values for which it is defined, typically within the reals but sometimes among the integers or complex numbers. For instance the natural domain of square root is the non-negative reals when considered as a real number function." This text passage is quite sloppy, by the way. What exactly is meant by "formula"? Does "square root" refer to $\sqrt{x}$ or rather to the formula $f(x) = \sqrt{x}$? Is it possible to speak of a natural domain e.g. of the equation $\sqrt{x-1}+\sqrt{x+1}=\sqrt{2x+\frac{3}{2}}$? Isheden (talk) 13:03, 23 April 2013 (UTC)
Firstly, a formula is a logical assertion. An identity is a formula, but an equation is not. $f(x) = \sqrt{x}$ is not a formula but the definition of the function f (supposing that it has not been defined previously; if it has been defined previously, the meaning depends on the context). The equation that you gives as an example is meaningful on every mathematical structure on which this expression is defined, and only on them. None of these structures is more natural than the others. For example, ax+b may refer to a number, to a polynomial in 3 variables, to a univariate polynomial with a and b as coefficients, to the function of x defined by this polynomial, to a vector and even to a line, if a and b are vectors and x a variable. In other word , one should not make a confusion between the syntax of an expression and its semantic which is provided by the context. When the IP user uses the word term, this refers to the syntax for constructing expressions. As such, there is no domain for a term. It may have a domain only when interpreted (semantic) implicitly of explicitly as a function. IMO, the notion of natural domain of a formula is WP:OR and also a mathematical nonsense for the above reasons. D.Lazard (talk) 14:07, 23 April 2013 (UTC)

## Name

Can someone explain why the name of this page is Domain of a function, and not Domain (Mathmetics)? FeyBart (talk) 18:51, 23 June 2013 (UTC)

Because the word domain can mean too many other things, even in mathematics. See domain#Mathematics for a (partial?) list. However, domain (mathematics) does currently redirect to this article, which muddies the waters a bit. --Trovatore (talk) 19:19, 23 June 2013 (UTC)