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A function is not really a function unless it's defined everywhere on its domain. So consider these three questions:

  1. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the square root function $f(x) = \sqrt{x}$. What is its domain?
  2. Let $f: [0, \infty) \rightarrow \mathbb{R}$ be the square root function $f(x) = \sqrt{x}$. What is its domain?
  3. Let $f$ be the square root function $f(x) = \sqrt{x}$. What is its domain?

Question 1 is ill-posed, because f is not a function.

Question 2 is trivial; the question gives you the answer.

Question 3 is ambiguous. The domain could be $[0,\infty)$ or $\mathbb{C}$ or something else.

So what is the best way to ask this question? I've just recently found out about partial functions but I'm not sure it would be a good idea to introduce this concept to my students.

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    $\begingroup$ perhaps, "Let $f:X \rightarrow \mathbb{R}$ be the square root function $f(x) = \sqrt{x}$ for $x \in X \subset \mathbb{R}$. What is $X$ (the domain)? or something to this end. The idea is to use a placeholder for the unknown --- in this case, the domain. $\endgroup$ – Math Misery Jun 28 '16 at 13:56
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    $\begingroup$ Various textbooks get at this problem by the notion of the "implied domain." It's not perfect, but a decent compromise. $\endgroup$ – user52817 Jun 28 '16 at 14:40
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    $\begingroup$ @MathMisery: That doesn't help, because $X$ could still be, say, $[5, 1729]$ or $\mathbb{N}$ instead of $[0, \infty)$. You have to state some criterion for what makes one choice of $X$ the "right" one. $\endgroup$ – Daniel Hast Jun 28 '16 at 16:33
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    $\begingroup$ The notation $f: A\to B$ implies that $A$ is the domain of function $f$. $\endgroup$ – Dan Christensen Jun 28 '16 at 18:32
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    $\begingroup$ @DanielHast sure, we can tighten this up a bit by requesting that $X$ be maximal ... again, being fast and loose in the comments, but something like "if $A$ is any other domain, then $A \subset X$" and we can further strict $X$ to be some subset of the reals [in the case of the square root example] if we wanted to avoid getting tangled with $\mathbb{C}$ $\endgroup$ – Math Misery Jun 30 '16 at 23:57
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It's not really a question about functions and domains, but about valid expressions. The question is, "For which real numbers $x$ is the following expression valid/well-defined/well-formed?" So, for example, $\frac{x + 1}{x^2 - 1}$ is valid exactly for $x \notin \{-1, 1\}$, and $\sqrt{x}$ (assuming this notation is established to mean the real-valued square root) is valid for all $x \geq 0$.

Bringing "functions" (which aren't actually functions unless the domain is already specified) into it is a distraction. Sure, you could try to formalize it as something like "the largest subset of the real numbers to which the formula for the function can be continuously extended", but this isn't what's really being asked: $\frac{x + 1}{x^2 - 1}$ isn't valid if $x = -1$, but can be continuously extended to all $x \neq 1$.

So, if you want to ask questions like this, recognize it as the syntactic question it is: the point of these questions isn't to work with functions themselves, but to become familiar with the notation used in mathematical expressions — and in particular, when it is and isn't valid/well-defined.

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  • $\begingroup$ Indeed, "Find the domain" usually really means (in the classroom context) "Find $\mathbb{R} \setminus X$, where $X$ is the set of points in $\mathbb{R}$ where the function can't be defined". It's odd that there isn't a standard word for $X$. $\endgroup$ – mweiss Jun 28 '16 at 16:43
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    $\begingroup$ @mweiss: As my example shows, it doesn't mean to exclude only the points where the function "can't be defined" (in the sense of being continuously extended or something), but where the given formula doesn't make sense (regardless of whether the resulting function could be extended further). It's really syntactic — the form of the expression matters, not just the values it takes for various $x$. $\endgroup$ – Daniel Hast Jun 28 '16 at 16:48
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    $\begingroup$ Yes, I see. My point is that there is no name for "the set where the formula doesn't make sense", so instead we appropriate the word "domain" and misuse it as if it means "the set where the formula does make sense". $\endgroup$ – mweiss Jun 28 '16 at 16:50
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user52817, in the comments, has this exactly right: In the context of Precalculus, the implied domain of a function is the largest subset of $\mathbb{R}$ on which the function is defined. So introduce that notion, and practice it with questions like "What is the implied domain of $f(x)=\sqrt{x}$?" (You can also let your students know that sometimes people are lazy and omit the word "implied", but there are good reasons not to do that.)

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    $\begingroup$ Your definition is circular, since you are assuming as given a function, which by definition has a domain, which is automatically the largest (and smallest) possible domain, since a different domain would, strictly speaking, give rise to a different function. You can circumvent this by asking "What is the largest possible domain for a well-defined function $f$ such that $f(x) = \sqrt{x}$ holds for all $x$ in the domain of $f$?" $\endgroup$ – Michael Joyce Jun 28 '16 at 17:16
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    $\begingroup$ This is the correct answer. The standard is certainly to introduce some advance definition so as to reduce the verbiage in these questions. I've also seen this expressed as an "Agreement on Domain" (Ratti & McWaters Precalculus, Sec. 2.4; Sullivan College Algebra Sec 3.1, etc.). $\endgroup$ – Daniel R. Collins Jun 28 '16 at 17:35
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Although there are good answers already, I think the ambiguity of the vocabulary is being underestimated. This might be because French definitions have more ambiguity than others, I can't tell.

In France, at least up to some years ago, the words function ("fonction") had various meaning depending on the context.

In high school (and probably some colleges), one would write let $f:\mathbb{R}\to\mathbb{R}$ be the function defined by $f(x)=\sqrt{x}$, because the first $\mathbb{R}$ would not mean the domain, but would be a mere notation for specifying the type of argument $f$ could be fed with. Then one can ask for the domain ("domaine de définition") of $f$, which is the largest subset of $\mathbb{R}$ for which the given expression makes sense.

At university, one would use the current mathematical convention for functions and domain, thus making more intricate to ask domain questions (but they are assumed to be well understood at that point -- not a very wise assumption unfortunately).

The vocabulary should be chosen so as to express easily what one wants to express, so while I have been uncomfortable with the above state of affairs, it has quite some ground. Take for example the case of unbounded operators in functional analysis: here the vocabulary matches quite closely the French high school vocabulary, and this is very useful.

I don't think it is either wise nor plausible to try have a completely formalized mathematical language: one will always abuse notation at some point (If I had more time I would dare anyone disagreeing to point to any mathematical text of at least a few pages that would be a counterexample, but I don't). If we could make students understand that $f$ has the type of a function and $f(x)$ has the type of a number, it would be a good thing. But we should allow ourselves and students some slack.

To finally answer the primary question, a reasonable phrasing could be:

Determine the largest real domain where a function $f$ can be defined by $f(x)=\sqrt{x}$.

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    $\begingroup$ I think that's the way to go. The largest (or maximal) domain makes a lot of sense. However, I would add a surrounding set, e.g. as "What is the largest set $D\subset\mathbb{R}$ where a function $f$ can be defined on $D$ by $f(x) = \sqrt{x}$?" $\endgroup$ – Dirk Jun 30 '16 at 8:08
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If you don't want to call $f$ a function, because it's not, call it what it is, a partial function:

Given the partial function $f : \mathbb R \to \mathbb R$ where $f(x) = \sqrt{x}$, what is the domain of $f$?

(This depends of course on what definition of domain you've selected for partial functions in your course. You might have to ask 'for what $x$ is $f(x)$ defined' instead. But you can avoid calling $f$ a function when it's not.)

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Really, the right way to go about things is

Find the set $X$ of points $x$ for which there exists a $y$ satisfying $y^2 = x$.

and then proceed to define $f(x) = \sqrt{x}$ for $x \in X$.

Similarly, for the example of $\frac{x+1}{x^2-1}$, the right question is

Find the set $X$ of points $x$ for which $x^2 - 1 \neq 0$

and only ever utter the expression $\frac{x+1}{x^2-1}$ when $x$ is restricted to $X$.

That said, I honestly believe that introductory materials that would ask questions like "what is the domain of this expression" like this are trying to teach partial functions, but are handicapped by trying to do so in the language of (total) functions.

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    $\begingroup$ You do need to say 'Find the set $X \subseteq \mathbb{R}$ of points $x$ for which there exists a $y \in \mathbb{R}$ such that $y^2 = x$. Otherwise a) you get a proper class (without restricting $x$) or b) you get all of $\mathbb{R}$ (without restricting $y$). $\endgroup$ – Jonathan Cast Nov 13 '17 at 16:24

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