jmc

Setting $x=1,$ we get $$f(2)+f\left(\frac{1}{2}\right)=1.$$Setting $x=-1,$ we get $$f\left(\frac{1}{2}\right)-f(2)=1.$$Subtracting these equations, we get $2f(2)=0,$ so $f(2)=\boxed{0}.$