When is $f^{-1}=1/f\,$?
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Russell Euler and James Foran. "On Functions Whose Inverse Is Their Reciprocal." Mathematics Magazine Vol. 54, No. 4 (Sep., 1981), pp. 185-189. JSTOR
If you're interested in complex solutions, the function $f(z) = z^i$ (using the principal branch) satisfies $f(f(z)) = 1/z$ for $e^{-\pi} < |z| < e^{\pi}$.