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 $f(x)=\sqrt{x}$.