  We show that if f: X --> Y is a finite, separable morphism of smooth curves
defined over a finite field F_q, where q is larger than an explicit constant
depending only on the degree of f and the genus of X, then f maps X(F_q)
surjectively onto Y(F_q) if and only if f maps X(F_q) injectively into Y(F_q).
Surprisingly, the bounds on q for these two implications have different orders
of magnitude. The main tools used in our proof are the Chebotarev density
theorem for covers of curves over finite fields, the Castelnuovo genus
inequality, and ideas from Galois theory.
