Given X = {x, y, z, w} and A = {a, b, c}, consider F: X $\rightarrow$ A and G: A $\rightarrow$ X, defined by:
$F(x) = a, F(y) = c, F(z) = c, F(w) = b$
$G(a) = y, G(b) = z, G(c) = w;$
I should be able to define $G◦F(w)$ and $F◦G(b)$ (which results, if I understood, are: z and c),a nd I should understand if $G◦F$ and $F◦G$ are injective and/or surjective. As far as I know, for example, $F◦G$ should be $(F◦G)(a) = F(G(a))$, $∀a$, and I know the definitions of injectivity and surjectivity, but I can't understand how I can study if the compositions are surjective, injective or bijective.
$F\circ G$ will be a function from $A \to A$. You should compute that function for all three elements of $A$. You have done one already. Having done that, you can just look to see if it is injective, surjective, or both. I find that it is neither. It is not surjective because there is no element that results in $a$ and it is not injective because $F\circ G(a)=F\circ G(b)=w$