Surjective Functions. Beginner in discrete Mathematics

Question

I am solving a problem related to surjective functions and I suspect that the question is wrong. The Question is as follows:

Let f : A → B and g : B → A be functions. Then prove that: If f and g are onto, then g ◦ f is one-to-one. What can you say about f ◦ g?

Following is my approach:

           Since f is onto, then for all 'b' belonging to B there is an 'a' belonging to A such 
           that f(a)=b.

           Also,

           Since g is onto, then for all 'a' belonging to A there is a 'b' belonging to B such that 
           g(b)=a.

           Now, to prove that gof is one-one, we need to show that if 
           gof(x1)=gof(x2) then x1=x2.

           Let us suppose gof(x1)=gof(x2),then
           g(f(x1))=g(f(x2))

           f(x1)=b1 where b1 belongs to B.
           f(x2)=b2 where b2 belongs to B.

           Therefore, g(b1)=g(b2)
                      g(b1)=a1 where a1 belongs to A
                      g(b2)=a2 where a2 belongs to A

           Therefore, a1=a2 (Which is not necessarily true)

Now , I am not able to proceed further. Is there anything I am doing wrong ?