enter image description herewe need to figure out the symmetry related to $$()+(1/x)=1$$ function? What symmetry should $ f(x)+f(1/x)=1$ follow? My teacher gave me this problem and I'm quite not sure how to solve this and looking for help here
This graph satisfies this very relation. Actually I want to know what symmetry this graph follows
For the simplicity assume that $f$ is a function form the set of no-zero reals $\Bbb R\setminus\{0\}$ to $\Bbb R$. Put $\Bbb R_1=\{x\in\Bbb R: |x|>1\}$. Then a family of all functions $f$ satisfying the given condition can be described as a family of all functions $f: \Bbb R\setminus\{0\}\to\Bbb R$ for which there exist a function $g:\Bbb R_1\to\Bbb R$ such that $f(x)=g(x)$ for each $x\in\Bbb R_1$, $f(1)=f(-1)=1/2$, and $f(x)=1-g(1/x)$ for each $x\in (-1,0)\cup (0,1)$.
This property of a function $f$ can be understood as a symmetry, but I think this is loose and does not correspond to the usual symmetries of functions from $\Bbb R$ to $\Bbb R$. For instance, there not necessary exists a non-identity motion of the real plane, which transforms the graph of the function $f$ to itself, as holds, for instance, for such symmetric functions as odd, even or periodic.