Let \(f:X \to Y\) and \(g:Y \to Z\) be functions. We can define the composition of \(f\) and \(g\) to be the function \(g\circ f:X \to Z\) for which the image of each \(x \in X\) is \(g(f(x))\text{.}\) That is, plug \(x\) into \(f\text{,}\) then plug the result into \(g\) (just like composition in algebra and calculus).
If \(f\) and \(g\) are both injective, must \(g\circ f\) be injective? Explain.
If \(f\) and \(g\) are both surjective, must \(g\circ f\) be surjective? Explain.
Suppose \(g\circ f\) is injective. What, if anything, can you say about \(f\) and \(g\text{?}\) Explain.
Suppose \(g\circ f\) is surjective. What, if anything, can you say about \(f\) and \(g\text{?}\) Explain.