2022 AMC 12B Problem 11

The two bases are the primitive cube roots of unity, $\omega = e^{\frac{2\pi i}{3}}$ and its conjugate $\omega^2 = e^{-\frac{2\pi i}{3}}.$ So $f(n) = \omega^n + \omega^{-n} = 2\cos\dfrac{2\pi n}{3}.$ Since $2022$ is a multiple of $3,$ $\omega^{2022} = 1,$ so $f(2022) = 1 + 1 = 2.$ Thus, the correct answer is E.