We propose a general method for describing tapping-mode atomic-force microscopy. The combined participation of attractive and repulsive interactions determines the multivalued nature of the resonance curve. This, in turn, implies the coexistence of two different stable oscillations for some excitation frequencies. The coexistence of two stable oscillations depends on the driving force and tip-surface separation. Increasing the driving force inhibits the low-amplitude oscillation state. Because resolution depends on the oscillation state, we propose that the absence of the low amplitude solution is responsible for the inconsistencies observed in high-resolution imaging of biomolecules.