An Application of the Hasse-Weil Bound to Rational Functions over Finite Fields

We use the Aubry-Perret bound for singular curves, a generalization of the Hasse-Weil bound, to prove the following curious result about rational functions over finite fields: Let $f(X),g(X)\in\Bbb F_q(X)\setminus\{0\}$ be such that $q$ is sufficiently large relative to $\text{deg}\, f$ and $\text{deg}\, g$, $f(\Bbb F_q)\subset g(\Bbb F_q\cup\{\infty\})$, and for ``most'' $a\in\Bbb F_q\cup\{\infty\}$, $|\{x\in \Bbb F_q:g(x)=g(a)\}|>(\text{deg}\, g)/2$. Then there exists $h(X)\in\Bbb F_q(X)$ such that $f(X)=g(h(X))$. A generalization to multivariate rational functions is also included.