  Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements, and let $m_1$
and $m_2$ be positive integers. Given polynomials $f_1(x), f_2(x) \in
\mathbb{F}_q[x]$ with $\textrm{deg}(f_i(x)) \leq m_i$, for $i = 1, 2$, and such
that the rational function $f_1(x)/f_2(x)$ belongs to a certain set which we
define, we present a sufficient condition for the existence of a primitive
element $\alpha \in \mathbb{F}_{q^n}$, normal over $\mathbb{F}_q$, such that
$f_1(\alpha)/f_2(\alpha)$ is also primitive.
