Cumulative velocity functions for galaxies and ΛCDM halos. The dashed black line represents the cumulative number density of halos as a function of their maximum circular velocity, V_h, in a ΛCDM universe (BolshoiP simulation; Hess et al., in prep.). The solid black line corresponds to the “galactic” HVF, which excludes massive halos that are unlikely to host individual galaxies (see Eq. (5)). The dotted green line shows the cumulative VF of ALFALFA galaxies. The red shaded region represents the VF of early-type galaxies, measured from SDSS data (Bernardi et al. 2010). The solid green line represents the VF of galaxies of all morphological types, obtained from the sum of the ALFALFA and early-type VFs.