  Let $K$ be a function field over an algebraically closed field $k$ of
characteristic $0$, let $\varphi\in K(z)$ be a rational function of degree at
least equal to $2$ for which there is no point at which $\varphi$ is totally
ramified, and let $\alpha\in K$. We show that for all but finitely many pairs
$(m,n)\in \mathbb{Z}_{\ge 0}\times \mathbb{N}$ there exists a place
$\mathfrak{p}$ of $K$ such that the point $\alpha$ has preperiod $m$ and
minimum period $n$ under the action of $\varphi$. This answers a conjecture
made by Ingram-Silverman and Faber-Granville. We prove a similar result, under
suitable modification, also when $\varphi$ has points where it is totally
ramified. We give several applications of our result, such as showing that for
any tuple $(c_1,\dots , c_{d-1})\in k^{n-1}$ and for almost all pairs
$(m_i,n_i)\in \mathbb{Z}_{\ge 0}\times \mathbb{N}$ for $i=1,\dots, d-1$, there
exists a polynomial $f\in k[z]$ of degree $d$ in normal form such that for each
$i=1,\dots, d-1$, the point $c_i$ has preperiod $m_i$ and minimum period $n_i$
under the action of $f$.
