Two classes of exceptional m-to-1 rational functions

For each odd prime power q, we present two rational functions \(f(X)\in \mathbb {F}_q(X)\) $f(X)\in {\mathbb{F}}_q(X)$ which have the unusual property that, for every odd n, the function induced by f(X) on \(\mathbb {F}_{q^n}\setminus \mathbb {F}_q\) ${\mathbb{F}}_{q^n}\backslash {\mathbb{F}}_q$ is \((q-1)\) $(q-1)$ -to-1.