We examine the nature of the central power source in very luminous inf
rared galaxies. The infrared properties of the galaxies, including the
ir far-infrared and 2.2 mu m fluxes, CO indices, and Brackett line flu
xes are compared to models of starburst stellar populations. Among sev
en galaxies we found two dominated by emission from young stars, two d
ominated by emission from an AGN, and three transition cases. Our resu
lts are consistent with evidence for active nuclei in the same galaxie
s at other wavelengths. Nuclear mass measurements obtained for the gal
axies indicate an initial mass function biased toward high-mass stars
in two galaxies. After demonstrating our methods in well-studied galax
ies, we define complete samples of high luminosity and ultraluminous g
alaxies. We find that the space density of embedded and unembedded qua
sars in the local universe is similar for objects of similar luminosit
y. If quasars evolve from embedded sources to optically prominent obje
cts, it appears that the lifetime of a quasar is no more than about 10
(8) yr.