AMC 12 2006a 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 $f$?

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