Is $ f \circ g $ invertible in the diagram below?

Question

I was working through Can the composition of two non-invertible functions be invertible? For the image below is $f \circ g$ invertible?

Thanks!

Answer 1

Yes, good job! You've correctly constructed a counterexample that demonstrates how $f \circ g$ can be invertible even though $f$ is not invertible (since $f$ only needs to be invertible when its domain is restricted to the range of $g$).

Answer 2

Your counterexample is good.

You can see in your example the two properties in action:

• If $f\circ g$ is surjective, then $f$ must be surjective (and yours is), but $g$ doesn't need to be surjective (and yours is not)
• If $f\circ g$ is injective, then $g$ must be injective (yours is), but $f$ does not need to be injective (yours is not)