Surjectivity of Composition of Surjective Functions Suppose we have two functions, $f:X\rightarrow Y$ and $g:Y\rightarrow Z$. If both of these functions are onto, how can we show that $g\circ f:X\rightarrow Z$ is also onto?

The important thing to learn when first studying mathematics is always to follow carefully and slowly with the definitions and theorems that you have seen in class.
Definition: Let $f\colon A\to B$, we say $f$ is surjective if for every $b\in B$ there is some $a\in A$ such that $f(a)=b$.
Definition: Let $f\colon A\to B$ and $g\colon B\to C$ functions, we define the composition $g\circ f$ as the function: $(g\circ f)(x) = g(f(x))$.
Of course it is perfectly possible that you were given different definitions in the course/book/etc. from which you study set theory. If indeed these are not the definitions you can try and prove the claim from the definitions you were given, or try to prove that the definitions above are the same.

Now suppose $f\colon X\to Y$ and $g\colon Y\to Z$ are two surjective functions, let $h$ be the composition, that is $h=g\circ f\colon X\to Z$. If we want to show that $h$ is surjective then we need to take an element $z\in Z$ and show that for some $x\in X$ we have $h(x)=z$.
Since we also know that $f,g$ are surjective we can pick some $y\in Y$ such that $g(y)=z$ and some $x\in X$ such that $f(x)=y$.
Now what can we say about $h(x)$?
