Let $f \colon A \to B$ and $g \colon B \to C$. Then
If $g \circ f$ is injective, then I know that $f$ is injective, but what can we say about the injectivity of $g$?
If $g \circ f$ is surjective, then I know that $g$ is surjective, but what can we say about the surjectivity of $f$?
On
No. For example,
(1) Let $A=\{1\}$, $B=\{4,5\}$ and $C=\{7\}$.
$f(1)=4$ and $g(4)=g(5)=7$. Clearly $g\circ f$ is injective and $g$ is not injective.
(2) Let $A=\{1,2,3\}$, $B=\{4,5,6\}$ and $C=\{7\}$.
$f(1)=f(2)=f(3)=4$ and $g(4)=g(5)=g(6)=7$. Clearly $g\circ f$ is surjective, however $f$ is not surjective.
No thing we can say about injectivity of $g$ except that it is injective on $f(A)$.
For next question also it means that $g$ maps $f(A)$ to entire $C$,no thing more.