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There are a number of questions recently which ask, "What is the domain of this function?" or something similar.

Strictly speaking the domain is part of the definition of any function in the first place. Actually "the" domain is indeterminate from the data given in such questions. The real question being asked tends to be something like, "What is the largest subset of the real numbers on which this formula defines a real-valued function?" (because of course the range often goes undescribed too, and if I add an infinity symbol to the range I can possibly change the domain...).

I guess it is just lazy shorthand, but is it essentially benign? Or are there real problems? Or am I wrong to be irritated?

Most such questions seem to inherit their defects from the source material, rather than being misunderstandings perpetrated by people asking questions.

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    $\begingroup$ For quite some time, I thought people ask questions because they don't understand quite a few things in mathematics. Reading your post, I begin to suspect I'm wrong: some may do that in order to irritate high-reputation members of this site. So we should consider adding something to our usual greetings: not just "What have you tried?" or "Use MathJax!", but also "Use proper terminology!". $\endgroup$ – Professor Vector Sep 11 '17 at 4:39
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    $\begingroup$ When I studied functions in school as part of set theory, it was always defined with two sets one domain and other co-domain (range being a subset of co-domain). But once we reach functions of real variables, these two sets are never mentioned explicitly and there are a slew of questions to find the largest possible domain and corresponding range. I think the primary reason for this is "almost" equivalence of a function $f$ with the expression for $f(x) $. The convention is not too harmful / confusing. $\endgroup$ – Paramanand Singh Sep 11 '17 at 6:46
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    $\begingroup$ Maybe I am confused, but I do not see what the problem is. I agree that a function has a definition that includes a domain (e.g., a function is a mapping from a first set to a second set such that each member of the first set maps to one and only one member of the second set, where the first set is a domain). In that case, it is grammatically correct to say "the domain of a function" because, by the definition, the function has a domain. $\endgroup$ – Ron Gordon Sep 12 '17 at 14:46
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    $\begingroup$ @RonGordon Suppose I have $f(x)=\frac 1{1+x}$, and I am asked what the domain of this function is. I could say that my choice of function has domain the positive reals. The intended answer is the reals excepting the real number $-1$. But I could consider this as a function on the Riemann Sphere too. I put it up for discussion because I feel that it may be over pedantic to be concerned about such things. However, identifying the implied domain in these cases makes more assumptions than the person asking the question normally realises. $\endgroup$ – Mark Bennet Sep 12 '17 at 19:12
  • $\begingroup$ Ok, now I see what the OP means. It is not merely a grammatical thing, it is a laziness thing on part of our educators. Good point! $\endgroup$ – Ron Gordon Sep 12 '17 at 21:59
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    $\begingroup$ I am curious: how should a good answer to this question look like? Remember that Meta.MSE shouldn't be used as a forum, despite the "discussion" tag. Anyway, the single truly irritating formulation on MSE is "solve this limit" (with variations: "solve this function", "solve this sequence" etc.). $\endgroup$ – Alex M. Sep 13 '17 at 15:01
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    $\begingroup$ @RonGordon "laziness thing on part of our educators" is not, I think, the same thing as "unwillingness on part of our educators to completely confuse students by being pedantic". (I mean, more US Calculus courses don't even teach limits with the $\delta$-$\epsilon$ definition at the first go-around) $\endgroup$ – Morgan Rodgers Sep 14 '17 at 16:48
  • $\begingroup$ For me, the domain of a map is the projection to its left component. When the word is used in some other sense, I'd have to think longer, but I don't think I'd be irritated. $\endgroup$ – user349106 Sep 14 '17 at 20:52
  • $\begingroup$ maybe they don't know image versus pre-image ? $\endgroup$ – user451844 Sep 17 '17 at 23:10
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    $\begingroup$ Would "find the implicit domain" irritate you less, or is it just as bad? $\endgroup$ – bof Sep 21 '17 at 20:44
  • $\begingroup$ @bof I think that would highlight the issue at hand. There is clearly an informal usage in common parlance. My question is really whether the informal usage is problematic. I am just (post this post but pre this comment) rereading GH Hardy's "Pure Mathematics" - which is ancient in terms of terminology, but careful about language. Richard Feynman was also careful. And I think this is a quality of the best educators. I wanted to highlight the issue not so much to be pedantic, but to work out how best to respond so that the people who pose such questions learn good mathematics. $\endgroup$ – Mark Bennet Sep 21 '17 at 21:20
  • $\begingroup$ Why does it have to be a subset of the real numbers? Why not complex numbers? ... or why not something other than numbers? E.g if asked what the domain of $f(x)=x$ is, it really could be anything $\endgroup$ – Bram28 Sep 22 '17 at 4:02
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    $\begingroup$ @Bram28 Of course it doesn't have to be the reals - but this is often the intended meaning given the wider context of the question. $\endgroup$ – Mark Bennet Sep 22 '17 at 7:08
  • $\begingroup$ @MarkBennet Hmm, but wouldn't that same context make it clear what is meant when the question 'what is the domain' is asked? And by the way, yes, I am someone else who is irritated by these 'what is the domain' questions ... in fact I have a whole big beef with the whole treatment of 'domain', 'co-domain', etc.: math.stackexchange.com/questions/2185073/… math.stackexchange.com/questions/1972560/… $\endgroup$ – Bram28 Sep 22 '17 at 14:01
  • $\begingroup$ @Bram28 Part of the context is that this is almost always evidently before considering complex variables. When complex variables are in view, poles become important points rather than excluded points, and the nature of singularities is thought of differently. $\endgroup$ – Mark Bennet Sep 22 '17 at 21:29
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You may be interested in this question on Math Educators: What is the proper way to ask a find the domain question.

It illustrates that it can be a bit tricky to ask this type of questions correctly, which is likely why not few resort to this phrasing. But, yes, I do consider it a bit of an abuse of terminology.

When faced with such a question I might point out that there is some issue with the phrasing, but would not insist on it too much, given that it is not unlikely that in the local context of the asker the phrasing is 'correct.'

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    $\begingroup$ It's extremely common for texts and instructors for (pre)calculus classes to use "domain" to mean something like "maximal subset of the reals for which the given formula is well-defined" (as described in the question you link). Like "closed", "domain" is just another word that has different conflicting meanings in different contexts. $\endgroup$ – Mark S. Sep 11 '17 at 23:51
  • $\begingroup$ As I said I do not massively object to that usage. A relevant question to me would be though whether these texts and instructors are at least internally consistent. Completely tangentially, what conflicting meanings of "closed" are there? $\endgroup$ – quid♦ Sep 12 '17 at 0:22
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    $\begingroup$ I think your answer is good, and intended my comment to add to it for anyone who feels as Mark Bennet does. Basically, a manifold (without boundary) is "closed" if its compact. But in other Hausdorff spaces, "compact" is a very special type of "closed". I suppose there's no serious conflict there since of course a whole manifold is literally closed, but it's always annoyed me a bit. $\endgroup$ – Mark S. Sep 12 '17 at 0:30
  • $\begingroup$ Thanks. The manifold usage of closed did not occur to me. $\endgroup$ – quid♦ Sep 12 '17 at 0:35
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    $\begingroup$ @MarkS. A better example might be the word "normal." How many different meanings does that little guy have? $\endgroup$ – Xander Henderson♦ Sep 17 '17 at 19:48
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Students pick this up from their textbooks. It is very common for algebra and pre-calculus, and calculus textbooks to present a function and ask students to find the domain of the function.

Students become accustomed to answering questions like "What is the domain of $f(x) = \frac{x-1}{x-2}$."

The students have no idea that some more learned people object.

(For what it's worth, some of us encourage the use of "the natural" or "conventional" or "implied" domain. But many of us do not write the textbooks.)

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    $\begingroup$ A more correct way to ask the question would probably be "For what $x$ is the expression defined?" $\endgroup$ – md2perpe Sep 12 '17 at 15:53
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    $\begingroup$ @md2perpe Yes, except in courses using textbooks that specify that the domains will consist only of real numbers. So the default domain in those courses is the maximum set of real numbers for which an expression is defined and real valued. (Examples include James Stewart's Calculus 8th ed and Earl Swokowski's 1st edition among others.) $\endgroup$ – Jim H Sep 12 '17 at 18:27
  • $\begingroup$ What would we say the domain of the function $\frac{\sin(x)}{x}$ is? $\endgroup$ – John Gowers Sep 14 '17 at 13:07
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    $\begingroup$ In an introductory calculus, we'd say it is all real numbers except $0$. $\endgroup$ – Jim H Sep 14 '17 at 18:21
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As you say, this type of question usually appears at a pre-calculus (i.e., high-school) level. It is clearly not formulated in the most rigorous way, but do you really think that is so important for those students?

I mean, I do not know what happens in your specific countries, but in mine many students say they do not like maths because it is useless (!) and too abstract. Clearly, 99% of the people in this world can live with that abusive notion of domain of a function, and probably that's the most intuitive definition for them (if it's intuitive at all).

A minor fraction of the students will continue their study of maths in college and, for those, the precise definition of domain of a function will certainly be introduced in their calculus courses.

Please don't be too picky.

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  • $\begingroup$ I think, from my experience, that the first way in which we encounter a concept is important in shaping how we deal with it. Seeing that concept in a different way can be really difficult. So using the same word informally and inaccurately can cause some people serious problems. Informal uses of words need not be inaccurate. But it seems to me that "domain" has acquired two different meanings. That is not necessarily a problem if the people who are using the word realise that it is ambiguous ("multi-valued") and make this explicit. But if the ambiguity is not addressed it can become a problem. $\endgroup$ – Mark Bennet Sep 14 '17 at 21:44
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    $\begingroup$ @MarkBennet From my experience (both as a teacher and a student) I don't agree that concept formation is a one-shot event. On the contrary, concepts are progressively grasped, when repeated again and again, from a variety of perspectives, whereas first time occurence is often unnoticed. Taking the argument to the extreme, we could not learn addition, since we do not know about the group structure, and we should first read a couple of volumes of Principia Mathematica... $\endgroup$ – Miguel Sep 16 '17 at 12:47
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    $\begingroup$ I must disagree rather strenuously with this answer. We should never tell students things that are not true. My goto examples are (1) the "white lie" that we tell small children about subtraction, namely that a larger number cannot be subtracted from a smaller number (it can be, but this requires negative numbers, which are beyond the scope of a typical kindergarten class; the right answer is "We don't yet know how to subtract bigger from smaller, so don't do it.), and (2) the square root of a negative number doesn't exist (it does, it is simply imaginary). (cont below). $\endgroup$ – Xander Henderson♦ Sep 17 '17 at 19:42
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    $\begingroup$ In the current context, giving an incorrect or hand-wavy definition of the domain likely won't cause much lasting harm, but the effort required to make the question rigorous also isn't that great. Instead of asking "What is the domain of $x\mapsto \frac{1}{x}$?", as "What is the largest set of real numbers that could be the domain of $x \mapsto \frac{1}{x}$?" It is correct, won't lead to misconceptions down the road, and students will get the idea just as quickly (i.e. after a few illuminating examples). $\endgroup$ – Xander Henderson♦ Sep 17 '17 at 19:45
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I posted this elsewhere, but I think it fits here as well.

One issue is that the abstract definition of a function as a left-total single-valued relation does not always match the use of the word "function" in the wild.

And elementary calculus, in particular, tends to use traditional terminology. Another example is the term "indeterminate form".

Recall that G.H. Hardy explained in his "Course in Pure Mathematics" (1908, 10th edition 1952) that being defined for every input and giving only one value for every input were "by no means involved in the general idea of a function" (his exact words, section 20). He went on to write, "All that is essential is that there should be some relation between $x$ and $y$ such that to some values of $x$ at any rate correspond values of $y$."

This kind of traditional view of a "function" is still alive in elementary calculus and in physics and other applied areas, even if it does not match the formal definition of a "function" in other areas of math.

Unfortunately, some books seem to mix the two notions: they talk about the modern interpretation of "function", but then give questions such as "find the domain of $\sqrt{3-x^4}$" that make more sense from the traditional interpretation.

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  • $\begingroup$ Perhaps a shadow of the old terminology shows up in function existence theorems like the implicit function theorem, which theoretically gives an open domain, but one could also attempt to determine the largest domain of definition. $\endgroup$ – Kyle Miller Sep 22 '17 at 17:12
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I understand the intent of this question, and I agree that simply handing a student a formula and asking for a domain is less than rigorous. However...

When such a question is asked on MSE, what is one to do? Does it help a precalculus student for us to talk to them about how their teacher or textbook isn't complying with a professional mathematician's standards of rigor, when that student needs to be learning which real numbers they can and can't plug in to the expression $\frac{x}{\sqrt{7-x}}$ in the context of a precalculus class?

The answers I sometimes see to these questions, critiquing the form of the question, seem singularly unhelpful to the student. Whether or not we find the form of the question irritating, surely the best reaction is to swallow that irritation and help lead the student to the understanding they need to succeed in their class?

I can see adding a note at the end of the answer that, in higher math, we would phrase such a request differently, for reasons, but to make that objection the central focus of the answer, as I've seen done on MSE, seems wrong-headed.

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There are a number of questions recently which ask, "What is the domain of this function?" or something similar. Strictly speaking the domain is part of the definition of any function in the first place.

There is no problem here whatsoever. Yes, the domain is part of the definition of a function, so the student is now asked to tell you what the domain is.

But, "definition" is a weasel word. The definition given to the student implicitly (and uniquely) defined the domain. So it is a little work to write down the set more explicitly. I feel this ambiguity is driving your question, but it doesn't diminish the rigor of the question or answer in the slightest.

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    $\begingroup$ The definition only defines the domain uniquely if the student has at some point been told that they always need to find the largest possible subset of the reals as domain. There are plenty of other reasonable options. $\endgroup$ – Tobias Kildetoft Sep 21 '17 at 6:39
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    $\begingroup$ @TobiasKildetoft must the students also be told that the answer they supply should be correct, and must they be told they should follow exactly the instruction that the answer they supply should be correct? $\endgroup$ – djechlin Sep 21 '17 at 18:33
  • $\begingroup$ @djechlin: Why would they need to be told that? That's quite different from excluding the possibility that the range or domain might be $\mathbb{C}$, or $\mathbb{R} \cup \left\{+\infty, -\infty\right\}$, or various other equally valid possibilities that give rise to different (but correct) answers? $\endgroup$ – psmears Sep 22 '17 at 10:11
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    $\begingroup$ @psmears These are all homework questions usually covered before complex numbers. "Domain" is well-defined for them at this point and we have no trouble working with this convention, except when we do, which is when we clarify. Higher level math is exactly the same. Sometimes rings have units sometimes they don't, it's a matter of communication and convention to know which you're dealing with. $\endgroup$ – djechlin Sep 22 '17 at 12:59
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I teach precalculus and trigonometry at a large public university, and we are specifically told to teach domain this way. That is, if a function is given by a formula, then the domain of the function is understood to be all values of $x$ for which the formula makes sense.

I don't personally like to think of a domain this way, but I think it would be more confusing for the students to define the domain any differently. Many people in this course have poor algebra skills and had a terrible high school math education. They could not possibly cope with such abstraction.

I say let the questions stand as they are. These students are struggling to get to the point where they can learn calculus. Explain things on their level, don't impose your views on rigorous mathematics.

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  • $\begingroup$ The domain is understood to be all values of $x$ for which the formula makes sense. Therein lies the problem. What is the domain of $\sqrt x$? Is it the non-negative real numbers? Why? If all I know about is integers, then the answer is the set $\{0, 1, 4, 9, ...\}$ If I know about Riemann surfaces, it's something else altogether. $\endgroup$ – mathguy Sep 23 '17 at 2:31
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    $\begingroup$ Precalculus students don't think of square root functions the same way as we do and they shouldn't have to. Instead of imposing our more nuanced perspective on domains of square root function on students, we make life easy for them and just say the square root of a negative number is undefined. $\endgroup$ – D_S Sep 23 '17 at 18:43
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I just offered this as a Comment to one of the Answers, but I realize it is in fact an Answer to the original question also. So I am posting it here.

In essence, I agree with the OP. It is true that the question is often asked at elementary levels, but it teaches bad habits. If pedantry is bad (perhaps it is), then don't ask about domains at all. How is asking about domains in an incorrect way less damaging than not asking about domains at all?

Anyway... here is my answer:

The domain is understood to be all values of $x$ for which the formula makes sense. Therein lies the problem. What is the domain of $\sqrt x$? Is it the non-negative real numbers? Why? If all I know about is integers, then the answer is the set $\{0, 1, 4, 9, ...\}$ If I know about Riemann surfaces, it's something else altogether. https://www.youtube.com/watch?v=mIOvmCyT4DQ

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  • $\begingroup$ "If pedantry is bad (perhaps it is), then don't ask about domains at all." Domain is an important thing for people doing basic math to know. But doing it "right" comes at a price. $\endgroup$ – D_S Sep 23 '17 at 18:49
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Given a partial function $f : X \rightarrow Y$, the set $$\{x \in X : f(x) \neq \bot\},$$ is sometimes referred to as the domain of definition of $f$.

However, this is very confusing terminology. For starters, the domain of definition, which is different from the domain, usually isn't stated in the definition; you have to find it. To make things worse, once you've found it, you can define a new function, whose domain is the domain of definition of the original function, and abusively denote this by the same letter as the partial function, and now the domain of definition has become a genuine domain, through an implicit act of definition. And to make things even worse, most books don't explain any of this, because for some reason, partial functions aren't taken seriously by most math educators. So, it's really no surprise that students are a bit confused about this, or at least unsure of how to express themselves.

In short, this isn't really the fault of students.

My recommendations for fixing the problem are:

  • Make the concept of partial functions explicit.
  • Explain that, for example, division of real numbers can be viewed as having domain/codomain $\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$, so it's domain is $\mathbb{R} \times \mathbb{R}$. However, since it's merely a partial function, evaluating it at a pair $(a,b)$ may return $\bot$ instead of a real number.
  • Choose a name for the domain of definition that sounds less like 'domain' and doesn't involve 'definition.' For example: pre-image, pre-range, co-range, any of these are fine. You could even call it the support, though the support of a function is usually taken to be a closed set, in order for conditions like "compactly-supported function" to be downward-closed.
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    $\begingroup$ -1 for suggesting that anything written here can be seen as standard in any way. $\endgroup$ – Tobias Kildetoft Sep 20 '17 at 5:32
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    $\begingroup$ @TobiasKildetoft, well, for starters, wikipedia disagrees with you. And -1 to you, for doing what many user's on this website do when confronted with partial functions and/or clever use of monads, namely: not doing your research, and attacking someone in your ignorance. It really is quite shameful. $\endgroup$ – goblin Sep 20 '17 at 5:58
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    $\begingroup$ You linked to the page about partial functions. Those are a non-standard object already, and suggesting that they are at all relevant here is just wrong. $\endgroup$ – Tobias Kildetoft Sep 20 '17 at 6:26

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