Functional equation: what function is its inverse's reciprocal? The fact that so many students confuse functional inverse notation
$$f^{-1}(x)$$
with multiplicative inverse notation
$$[f(x)]^{-1}$$
got me to thinking... does there exist a function whose inverse is its inverse? That is, is there a function $f:\mathbb R_+\mapsto \mathbb R_+$ whose functional inverse is also its multiplicative inverse, so that
$$f^{-1}(x)=[f(x)]^{-1}, \space\space\space \forall x\in\mathbb R_+$$
Any ideas? I'll impose the restriction of continuity to deter nasty solutions.

Call $g=\ln\circ f\circ \exp:\Bbb R\to\Bbb R$. Then, $g^{-1}=\ln\circ f^{-1}\circ \exp=\ln\frac1{f\circ \exp}=-g$.
So, we want the homeomorphisms $g:\Bbb R\to\Bbb R$ such that $g^{-1}=-g$. But the homeomorphism $\Bbb R\to\Bbb R$ are strictly monotone continuous functions. And, if $g$ is strictly monotone, $g^{-1}$ must be monotone of the same sign. This is not consistent with $g^{-1}=-g$. Thus, there is no such $g$ and no such $f$.
