I am reading Quantum Computation and Quantum Information by Chuang and Nielsen and they claim that it is easy to show that transformation $U_f: \left| x, y \right\rangle \to \left| x, y \oplus f(x) \right\rangle$ is unitary.
My first instinct is to use the condition $U^\dagger U = I$, but I can’t figure out how to construct the matrix $U_f$. Is there another way?