Given that \(f(x)\) is continuous and differentiable for \(x \neq 0\), that \(f(-1) = 1\), and that \begin{align} (f(x))^2 + (f(y))^2 = f(x^2 + y^2) \quad \text{for all real } x, y, \end{align} show that \(f(x) = |x|\).