Showing that f,g are invertible if $A$ is a finite set and $f,g: A\to A$ such that $f\circ g$ is invertible

Question

Let $A$ be a finite set and $f,g: A\to A$ such that $f\circ g$ is invertible.

1. Prove f,g are invertible.

2. Prove that if $A$ is an infinite set, it doesn't mean that f,g are invertible.

I get that since $f\circ g$ is a bijection, then $g$ is an injection, $f$ is a surjection. What's left is to show that $f$ is an injection, $g$ is a surjection.

Now I just don't know what to with the fact that $A$ is a finite set...

Answer 1

Hint: a function $A \to A$ from a finite set $A$ to itself is injective if and only if it is surjective.

For 2., let us try $A = \mathbb{Z}_{\geq 0}$. You want $f$ to be surjective but not injective. An example would be $f: x \mapsto \lfloor x/2 \rfloor$. Now $g: x \mapsto 2x$ is injective but not surjective... What is $f \circ g$?