Finding functions $f: \Bbb R_*^+ \to \Bbb R_*^+$ with certain properties

Question

Find the functions $f:\Bbb R_*^+ \to \Bbb R_*^+$ such that:

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

Answer 1

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

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

Answer 2

For any function $\tilde{f}:[1,\infty)\rightarrow \mathbb{R}_+^*$ such that $f(1)=1$, there is a unique extension $f:(0,\infty)\rightarrow \mathbb{R}_+^*$ defined for $x<1$ by $f(x)=1/\tilde{f}(1/x)$.