Let $p_1, ..., p_n$ be $n$ distinct points in $\mathbb{C}$, and let $q_1, ..., q_n$ be $n$ distinct points in $\mathbb{C}$. Is there a rational function $f \in \mathbb{C} (x)$ such that $f(p_i) = q_i$ for each $1 \leq i \leq n$? Thinking of $\mathbb{C}(x)$ as a monoid under composition, this question asks whether $\mathbb{C}(x)$ acts $n$-transitively on $\mathbb{C}$.
More importantly, if this is false, then is there a nice description of the set of pairs $((a_1, ..., a_n), (b_1, ..., b_n)) \in \mathbb{C}^n \times \mathbb{C}^n$ for which there is a rational function $f \in \mathbb{C}(x)$ such that $f(a_i) = b_i$?
A rational function is determined by its preimages counting multiplicities at 3 distinct points, so I suspect this is false.
By Lagrange interpolation, you don't even need rational functions — polynomials suffice.