How can I find all involutions whose reciprocals are also involutions?

Question

How can I solve the following functional equation?

$$\frac{1}{f(x)} = f\left( \frac{1}{x} \right)$$

This functional equation amounts to finding all involutions whose reciprocals are also involutions.

I've found two solutions, $y=\pm\frac{1}{x}.$

Answer 1

$\dfrac{1}{f(x)}=f\left(\dfrac{1}{x}\right)$

$f(x)f\left(\dfrac{1}{x}\right)=1$

In fact this functional equation belongs to the form of http://eqworld.ipmnet.ru/en/solutions/fe/fe2111.pdf.

$\therefore f(x)=\pm e^{C\left(x,\frac{1}{x}\right)}$ , where $C(u,v)$ is any antisymmetric function.