The wave function of a Tonks-Girardeau (T-G) gas of strongly interacting bosons in one dimension maps onto the absolute value of the wave function of a noninteracting Fermi gas. Although this fermionization makes many aspects of the two gases identical, their equilibrium momentum distributions are quite different. We observed dynamical fermionization, where the momentum distribution of a T-G gas evolves from bosonic to fermionic after its axial confinement is removed. The asymptotic momentum distribution after expansion in one dimension is the distribution of rapidities, which are the conserved quantities associated with many-body integrable systems. Our measurements agree well with T-G gas theory. We also studied momentum evolution after the trap depth is suddenly changed to a new nonzero value, and we observed the theoretically predicted bosonic-fermionic oscillations.