The galaxy mass function, i.e., the distribution of galaxies as a function of mass, is a useful way to characterize the galaxy population. In this work, we examine the stellar and baryonic mass function, and the velocity function of galaxies and galaxy groups for two volume-limited surveys of the nearby universe. Stellar masses are estimated from multi-band photometry, and we add cold atomic gas from measurements and a newly calibrated estimator to obtain baryonic mass. Velocities are measured from the internal motions of galaxies and groups and account for all matter within the system. We compare our observed mass and velocity functions with the halo mass function from theoretical simulations of dark matter, which predict a much more steeply rising low-mass slope than is normally observed for the galaxy mass function. We show that taking into account the cold gas mass, which dominates the directly detectable mass of low-mass galaxies, steepens the low-mass slope of the galaxy mass function. The low- mass slope of the baryonic mass function, however, is still much shallower than that of the halo mass function. The discrepancy in low-mass slope persists when examining the velocity function, which accounts for all matter in galaxies (detectable or not), suggesting that some mechanism must reduce the mass in halos or destroy them completely. We investigate the role of environment by performing group finding and examining the mass and velocity functions as a function of group halo mass. Broken down by halo mass regime, we find dips and varying low-mass slopes in the mass and velocity functions, suggesting that group formation processes such as merging and stripping, which destroy and lower the mass of low-mass satellites respectively, potentially contribute to the discrepancy in low-mass slope. In particular, we focus on the nascent group regime, groups of mass ∼10^11.4−10^12 Msun with few members, which has a depressed and flat low-mass slope in the galaxy mass and velocity function. We find that nascent groups are at the peak baryonic collapse efficiency (group-integrated cold baryonic mass divided by the group halo mass), while isolated dwarfs in lower mass halos are rapidly growing in their collapsed baryonic mass and larger groups are increasingly dominated by their hot halo gas. Scatter in this collapsed baryon efficiency could indicate varying hot gas fractions in nascent groups, suggestive of a wide variety of group formation processes occurring at these scales. We point to this nascent group regime as a period of transition in group evolution, where merging and stripping remove galaxies from the population, contributing to the discrepancy in low-mass slope between observations and dark matter simulations.