Is f 1-1? Suppose f(a, b) = f(c, d). Must (a, b) = (c, d)? If yes, then f is 1-1. If no, then f is not 1-1.
Is f surjective? Pick some element in Q, say a/b. Is there a (p,q) such that f(p,q) = a/b. If yes, then f is surjective. If not then f is not surjective.
Is f a bijection? What is the definition of a bijection?