Prove that, if f:A->B and g:B->C are functions with gof:A->C bijective, then f is injective and g is surjective.

Now, I easily saw how g is surjective:

Let x be an element of A

Since gof:A->C is bijective, g(f(x)) is bijective

Implies g(f(x)) is surjective and g(f(x)) is injective

Thus g is surjective.

But I've been fumbling around with trying to show that f is injective and I get lost. I'm sure it's something stupid I'm missing. Can anyone start me off in the right direction?