The technical, and somewhat archaic
answer, is that all
knowledge and inquiry is the realm of philosophy
. That is, everything that is now a science or a department at a university was once a part of philosophy, and still is, in a way. Science
-- all of it -- was once called "natural philosophy
." Almost all the greatest mathematicians in most of the history of mathematics have "also" been what we would today call philosophers
And today the term is used differently. As it is used in universities, philosophy usually means all those questions that have yet to be "farmed out" to one of the "sub-disciplines." There are no inherently philosophical questions -- just questions that are still philosophical. "Why do I prefer to comb my hair to the left?" (rimrod's example above) has been a philosophical type question for a while, but psychology and cognitive science are arguably ready to take it on. And even the question "Why do I have brown hair" could, before genetics, have been discussed as a philosophical one.
Chomsky makes a handy distinction bewteen problems and mysteries: Problems are questions for which we know what an answer would look like. (ie How does the human genome work?) Mysteries are questions for which we still have no idea what the answer would look like. (ie the Mind/Body problem, which by this terminology is actually the Mind/Body mystery). One role of philosophy, as I see it, is to turn mysteries into problems, and guide them on their way to being answered.
Now, there are some things that philosophy deals with that just don't seem to be amenable to being turned into problems. Ethics is a good example -- can we have a science of ethics? Or how about that wonderful question -- "How shall we live?" As well, some issues come back to philosophy for seconds: in cognitive science, for instance, philosophers are on call all the time, for diving in to mysteries like language and thought.
Because of its nature, the issue of its own realm is itself a philosophical question. We reserve the right to change it at will.