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f(f(x)) = − x, Windmills, and Beyond

Martin Griffiths
Mathematics Magazine
Vol. 83, No. 1 (February 2010), pp. 15-23
DOI: 10.4169/002557010x479956
Stable URL: http://www.jstor.org/stable/10.4169/002557010x479956
Page Count: 9
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Article
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f(f(x)) = − x, Windmills, and Beyond
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Abstract

Summary 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.

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