Suppose I have self-maps $g$ and $h$ from set $X$ to $X$ such that $g$ $\circ$ $h$ is a constant function and $h$ is surjective. What can be said about function $g$?
I think that g should also be a constant function, but I do not know how to prove that? Any help or tips?
Yes, $g$ is constant. For any $x\in X$, there exists $y\in X$ with $h(y)=x$, since $h$ is surjective. But we know that $g(x)=g(h(y))=c$ where $c$ is the appropriate constant. Since this is true for all $x$, we have that $g$ is constant.