Suppose we have functions $f:A→B$ and $g:B→C$. Prove that if $f$ and $g$ are invertible, then so is $g \circ f$.

Question

Suppose we have functions $f:A→B$ and $g:B→C$. Prove that if $f$ and $g$ are invertible, then so is $g \circ f$. Is the converse true? I.e., if $g \circ f$ is invertible, does it follow that $f$ and $g$ are both invertible?

I know that if $f$ and $g$ are invertible, then they are both injective and surjective. Therefore, I assume from this that I have to show $g \circ f$ is injective and surjective as well, thus proving it is invertible. However, I don't know how to show $g \circ f$ is bijective based only on the information I have.

Answer 1

Hint: You dont need to show that $gof$ is bijective.Instead,show that $f^{-1} \circ g^{-1}=(g\circ f)^{-1}$.Hence $g\circ f$ is invertible.

Answer 2

To help with part of your question:

The composition could be invertible, with the individual functions not being so. See if you can find an example with $A$ having $2$ elements, $B$ having $3$ elements, and $C$ having $2$ elements.