A question from a previous qualifying exam at my university reads:
"Suppose that f and g are entire functions such that $f \circ g(x) = x$
when $x \in \mathbb{R}$. Show that $f$ and $g$ are linear functions."
One can conclude that the composition of $f$ and $g$ is the identity on all of $\mathbb{C}$, by the uniqueness principle. I know how to solve the problem if one assumes that $f$ is injective. However, there are examples of functions that have a right inverse but are not injective. However, entire functions have many properties, so is there a way of showing $f$ must be injective from the information above, or should I approach the problem differently?
Best Answer
Injectivity of $f:$ Note that $$(g\circ f) (g(z)) = g[(f\circ g)(z)] = g(z).$$ Thus $g\circ f$ is the identity on $g(\mathbb C).$ Since $g(\mathbb C)$ is a set with limit point, the identity principle shows $g\circ f$ is the identity on $\mathbb C.$ This proves $f$ is injective.