Does there exists an irreducible polynomial of a given form?

Question

Let $q$ be a prime power, $n$ a positive integer, and $f\in \mathbb F_q[x]$ be an irreducible polynomial. Does there exists $g \in \mathbb F_q[x]$ of degree $n$ such that $f(g)$ is irreducible? I can deal with the case $n|q-1$, the case $n=q=p$ prime, and the case in which $q^{\deg(f)}$ is much larger than $n$.