Suppose we have functions f: AB and g: BC. Define the composition of f and g to be the function gf : A C such that (g f)(x) = g(f(x)).

(a) Prove the following claims: if g f is one to one (injective), then f is also one to one.

(b) Prove that g is not necessarily one to one, even if g f is.

(c) Prove that, if g f is onto (surjective), then g is also onto.

(d) Prove that f is not necessarily one to one, even if g f is.