By recourse to tempered ultradistributions, we show here that the effect of a q-Fourier transform (qFT) is to map equivalence classes of functions into other classes in a one-to-one fashion. This suggests that Tsallis' q-statistics may revolve around equivalence classes of distributions and not individual ones, as orthodox statistics does. We solve here the qFT's non-invertibility issue, but discover a problem that remains open.Facultad de Ciencias Exactas