This paper is concerned with the effect of damage on free and forced vibrations of a functionally graded cantilever beam. The modes of damage considered in the paper include a region with degraded stiffness adjacent to the root of the beam, a single delamination crack, and a single crack at the root cross section of the beam propagating in the thickness direction. Closed-form solutions are suggested for all cases considered, including both forced and free vibrations; in the case of free vibrations, these solutions are exact. The peculiarities of the frequency analysis of nonprismatic and/or axially graded beams with the root crack in the presence of static thermal loads are also discussed and it is shown that neglecting axial inertia may lead to a qualitative error (this conclusion remains valid in prismatic functionally graded material beams). Numerical examples concentrate on the effect of a single root crack on the fundamental frequency, because such damage was observed in numerous loading scenarios. It is shown that the presence of a crack that has propagated through about one-third of the thickness of the beam significantly affects the fundamental frequency.