- The domain of
f·g is the set of all x in the domain of g
for which g(x) is in the domain of f.
This is mathematical English, not colloquial English. In colloquial English for which and such that cannot be freely interchanged; but in mathematical phrasing like this, they can. This sentence means the same thing:
- The domain of
f·g is the set of all x in the domain of g
such that g(x) is in the domain of f.
For which introduces a restrictive relative clause bound by the quantifier all; it restricts the set to only certain x's, and the property those x's share is stated the set phrase for which (or such that).
These set phrases are used when a simple relative clause won't do the job. If the sentence was
- The domain of
P is the set of all x in Q
which are zeroes of the Zeta function.
there's no need to pied-pipe a preposition for, and there's no problem.
But in the example provided, all x isn't the subject of the restrictive relative,
so you can't use which as the subject, but rather posit some oblique relation,
and that calls for a preposition, which gets stranded when which is moved to the front
- *The domain of
f·g is the set of all x in the domain of g
which g(x) is in the domain of f for.
which is so clumsy that it's promptly pied-piped and glued together with which into a fixed phrase.
