This talk will survey methods of solving for functions implicitly defined by relationships between their iterates. We introduce general notions of implicitly defined functions and functional equations, including finite difference relations, Cauchy’s equation, Schroeder’s equation and iterative functional equations. This talk will focus on iterative functional equations and methods for solving them, including eigenvalue solutions, fixed-point iteration, and graphical representation. In particular we discuss graphical representations of endofunctions, efficient data structures for representing them and the insights they provide about the behavior of functions under iteration. Emphasis will be on introduction to the topic via examples, and ideas for pursuing further interest in the subject.