Are rational functions the only functions that are preserved by field automorphisms?

Question

I am reading Allufi's Algebra: Chapter 0 and I tried to think about what makes polynomials so important to field extensions. I realised part of the reason is the fact that if $L/K$ is a field extension, $f(x) \in K[x]$ is a polynomial and $\varphi \in Aut_K L$ is a K-automorphism of L, then $\varphi (f(x))=f( \varphi (x))$, i.e. polynomials are preserved by K-automorphisms. Obviously, rational functions are another type of function for which it is true, but I'm wondering whether those are the only possibilities, or are there any other functions which are preserved by K-automorphisms.