Let $K = \mathbb F_p(x)$, and let $H = \left\{\begin{pmatrix} d & a \\ 0 & 1 \end{pmatrix} \ \big| \ a \in \mathbb F_p, d \in \mathbb F_p^\times\right\}$ be a group under multiplication which acts on $K$ via $\begin{pmatrix} d & a \\ 0 & 1 \end{pmatrix} x = dx + a$.
How can I find the fixed field of $H$?
If $f = \frac{g(x^p - x)}{h(x^p - x)}$, then it's image under a generic point is $\frac{g(d(x^p-x))}{g(d(x^p-x))}$, which is close but isn't quite there.
Any help would be appreciated. Thanks