  Let $R(x)=g(x)/h(x)$ be a rational expression of degree three over the finite
field $\mathbb{F}_q$. We count the irreducible polynomials in
$\mathbb{F}_q[x]$, of a given degree, which have the form $h(x)^{\mathrm{deg}\,
f}\cdot f\bigl(R(x)\bigr)$ for some $f(x)\in\mathbb{F}_q[x]$. As an
application, we recover the number of irreducible transformation shift
registers of order three, previously computed by Jiang and Yang.
