I was wondering is there are any functions which are their own inverse and which consist of another function added with its inverse. In essence, I am interested in solutions for
$$G^{-1}(x) = G(x) = k\cdot\left[f(x)+f^{-1}(x)\right]$$
I know this is true if $f(x)=f^{-1}(x)$ and $k=1/2$, but then $G(x)=f(x)$, which I find a little boring. Are there any functions that fit this condition where $f(x)\neq f^{-1}(x)?$
I experimented around a bunch, and I think I can construct a function like this when I just form the average of two inversable $f(x)$ and $f^{-1}(x)$ over and over again, like this
$$u_1(x) = \frac{1}{2}\left[f(x)+f^{-1}(x)\right]$$ $$u_2(x) = \frac{1}{2}\left[u_1(x)+u_1^{-1}(x)\right]$$ $$...$$ $$u_n(x) = \frac{1}{2}\left[u_{n-1}(x)+u_{n-1}^{-1}(x)\right]$$
Right now I have only looked at positive continous functions of real positive $x$. But I have no idea if this actually works.