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We demonstrate that the quantum phase transition (QPT) of the Rabi model and critical dynamics near the QPT can be probed in the setup of a single trapped ion. We first demonstrate that there exists equilibrium and nonequilibrium scaling functions of the Rabi model by finding a proper rescaling of the system parameters and observables, and show that those scaling functions are representative of the universality class to which the Rabi model belongs. We then propose a scheme that can faithfully realize the Rabi model in the limit of a large ratio of the effective atomic transition frequency to the oscillator frequency using a single trapped ion and, therefore, the QPT. It is demonstrated that the predicted universal functions can indeed be observed based on our scheme. Finally, the effects of realistic noise sources on probing the universal functions in experiments are examined.