Solutions

a) Let $x,y\in\operatorname{Dom}(g)$ such that $x\ne y$. Then, if $g(x)=g(y)$, we have

$$x=f(g(x))=f(g(y))=y,$$

which contradicts $x\ne y$.

b) Let $b$ be any number. Let $a=g(b)$. It then follows that

$$b=f(g(b))=f(a).$$

Note that we suppose that $b$ is in the domain of $g$ and that $g(b)$ is in the domain of $f$.