problem: Suppose that $f$ is an invertible function from $(-\infty,\infty)$ to $(-\infty,\infty)$.  If $f(f(x))=x$, find $f(x)-f^{-1}(x)$.
solution: Applying $f^{-1}$ to both sides of the equation $f(f(x)) = x$, we get $f^{-1}(f(f(x))) = f^{-1}(x)$.  By definition of the inverse function, $f^{-1}(f(x)) = x$, so $f^{-1}(f(f(x))) = f(x)$.  Then $f(x) = f^{-1}(x)$, so $f(x) - f^{-1}(x) = \boxed{0}$.