smc

We know that $f(x)+f\left(\frac{1}{x}\right)=x.$ Plugging in $x=\frac{1}{x}$ we get $$f\left(\frac{1}{x}\right)+f\left(\frac{1}{\frac{1}{x}}\right)=\frac{1}{x}$$ $$f\left(\frac{1}{x}\right)+f(x)=\frac{1}{x}.$$ Also notice $$f\left(\frac{1}{x}\right)+f(x)=x$$ by the commutative property(this is the same as the equation given in the problem. We are just rearranging). So we can set $\frac{1}{x}=x$ which gives us $x=\pm 1$ which is answer option $\boxed{\{-1,1\}}.$