Rational transformations over finite fields that are never irreducible

Rational transformations play an important role in the construction of irreducible polynomials over finite fields. Usually, the methods involve fixing a rational function Q and deriving conditions on polynomials F is an element of Fq[x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\in \mathbb {F}_q[x]$$\end{document} such that the rational transformation of F with Q is irreducible. Here we want to change the perspective and study rational functions with which the rational transformation never yields irreducible polynomials. We show that if the rational function is contained in certain subfields of Fq(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q(x)$$\end{document} then the rational transformation with it is always reducible. This extends the list of known examples.