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f(x)- f(y) = f(x)f(1/y)- f(y)f(1/x), periodic

Source: Switzerland - Swiss TST 1998 p1

February 19, 2020
functionalperiodicfunctionalgebra

Problem Statement

A function f:R−{0}→Rf : R -\{0\} \to R has the following properties: (i) f(x)−f(y)=f(x)f(1y)−f(y)f(1x)f(x)- f(y) = f(x)f\left(\frac{1}{y}\right)- f(y)f\left(\frac{1}{x}\right) for all x,y≠0x,y \ne 0, (ii) ff takes the value 12\frac12 at least once. Determine f(−1)f(-1). Prove that ff is a periodic function
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