The primary goal of this document is to record the asymptotic effects that preimage constraints impose upon the sizes of the iterated images of a random function. Specifically, given a subset P⊆Z_≥ 0 and a finite set S of size n, choose a function uniformly from the set of functions f:S→ S that satisfy the condition that |f^-1(x)|∈P for each x∈ S, and ask what |f^k(S)| looks like as n goes to infinity. The robust theory of singularity analysis allows one to completely answer this question if

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... accepts that 0∈P, that P contains an element bigger than 1, and that (P)=1; only the third of these conditions is a meaningful restriction. The secondary goal of this paper is to record much of the background necessary to achieve the primary goal.