Resumen de f(f(x))= -x, Windmills and Beyond

Martin Griffiths

• In this article we consider the solutions of a particular functional equation and some generalizations. This equation possesses non-trivial yet accessible solutions, and is rather appealing because of the way that the enforced graphical symmetry of any solution implies that it must possess a certain non-obvious analytic property. Indeed, a central theme here is this interplay of the visual with the analytic aspects of the problem. After giving a brief historical overview of functional equations, we go on to study the general properties shared by functions satisfying our particular equation, as well as specific solutions. Finally, solutions to a generalization of the equation are obtained.