How to prove that a composite function $f\circ g$ is bijective$?$
because i have two questions. if $f$ is injective and $g$ is surjective, can $g\circ f$ be both injective and surjective? because the question assumes both sets to be the same criteria ( $f$ injective, $g$ surjective) , so one question is whether $g\circ f$ is injective and the other is if $g\circ f$ is surjective.
i proved that if $f$ is injective then, $x=y$, $f(x)=f(y)$, then $g(f(x)) = g (f(y))$ so $g\circ f$ is injective. but is it true that if $f$ is injective and $g$ is surjective, then $g\circ f$ can also be surjective. its making me confused.