Let K be a function field over an algebraically closed field k of characteristic 0, let ϕ ∈ K(z) be a rational function of degree at least equal to 2 for which there is no point at which ϕ is totally ramified and let α ∈ K. We show that for all but finitely many pairs (m, n) ∈
$\mathbb{Z}$
⩾0 ×
$\mathbb{N}$
there exists a place
$\mathfrak{p}$
of K such that the point α has preperiod m and minimum period n under the action of ϕ. This answers a conjecture made by Ingram–Silverman [13] and Faber–Granville [8]. We prove a similar result, under suitable modification, also when ϕ has points where it is totally ramified. We give several applications of our result, such as showing that for any tuple (c
0, . . ., c
d−2) ∈ k
d−1 and for almost all pairs (mi
, ni
) ∈
$\mathbb{Z}$
⩾0 ×
$\mathbb{N}$
for i = 0, . . ., d − 2, there exists a polynomial f ∈ k[z] of degree d in normal form such that for each i = 0, . . ., d − 2, the point ci
has preperiod mi
and minimum period ni
under the action of f.