The objective of this research is to develop a method of ﬁnding a function which produces a given escape diagram. An escape diagram is a graphical representation of the set of inputs in the domain of a function which remain bounded under iteration of the function. This research considers functions which have domains that are subsets of the complex numbers, although the functions are not necessarily complex analytic. The ﬁndings of this research include two methods: the ﬁxed point method, and the conjugate method. The ﬁxed point method provides a set of conditions which are contingent on the boundary of the given escape diagram consisting of repulsive ﬁxed points (the boundary must be given as two functions deﬁned on the interval (y1,y2)). If a function satisﬁes the conditions, the function will produce the given escape diagram. The conjugate method uses conjugacy to modify functions which have already known escape diagrams. The resulting function produces an escape diagram which is a distortion of the escape diagram of the known function. These methods were tested numerically, using computer programs to determine whether the functions that they result in produce the correct escape diagrams for multiple cases. Both of these methods were found to be successful in the cases tested.