Injective and Surjective

I have managed to come up with a proof, but not sure if it's right.
It starts like this:
let f: A-->B and g:B--C, then gf:A-->C

since gf is surjective we know that for each c in C, there is an element a in A such that gf(a)=c

Yes, if $f: A \to B$ and $g: B \to C$ are functions then $g \circ f: A \to C$ is surjective only if g is surjective.

No, f does not need to be surjective.

See if you can find an example where g o f is surjective, g is surjective but f is not.

The example of g(x)=x^2 and f(x)= squareroot of x. Is that an acceptable example?

Well, now we need to be very careful! What are the domains of f and g in your example?

EDIT: actually, what are the domains and ranges? Your example will sink or swim depending on the answer to those two questions.

for g its g: R -> R. Just to be clear since I can't write it like the standard notation R is the set of real numbers.
f: R+ ->R. R+ is just positive real numbers.

In that case, no. $g$ isn't surjective, because it doesn't hit everything in R. Try changing the ranges/domains and you should (assuming I'm not half asleep) be left with a working example...

P.S. Does anyone know how to do blackboard bold on the forum LaTeX? The usual mathbb doesn't work...

Ah ok so if the range was positive real numbers and domain is the set of all real numbers, would that be ok?