2006 AMC 12A Problem 18

The function f has the property that for each real number x in its domain, 1/x is also in its domain and

f(x)+f\left(\frac{1}{x}\right)=x

What is the largest set of real numbers that can be in the domain of f?

\mathrm{(A) \ } \{x|x\ne 0\}\qquad \mathrm{(B) \ } \{x|x<0\}\qquad \mathrm{(C) \ } \{x|x>0\}\qquad \mathrm{(D) \ } \{x|x\ne -1\; \mathrm{and}\; x\ne 0\;\mathrm{and}\; x\ne 1\}\qquad \mathrm{(E) \ } \{-1,1\}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.