  The study of ultracold optically trapped atoms has opened new vistas in the
physics of correlated quantum systems. Much attention has now turned to
mixtures of bosonic and fermionic atoms. A central puzzle is the disagreement
between the experimental observation of a reduced bosonic visibility ${\cal
V}_b$, and quantum Monte Carlo (QMC) calculations which show ${\cal V}_b$
increasing. In this paper, we present QMC simulations which evaluate the
density profiles and ${\cal V}_b$ of mixtures of bosons and fermions in
one-dimensional optical lattices. We resolve the discrepancy between theory and
experiment by identifying parameter regimes where ${\cal V}_b$ is reduced, and
where it is increased. We present a simple qualitative picture of the different
response to the fermion admixture in terms of the superfluid and
Mott-insulating domains before and after the fermions are included. Finally, we
show that ${\cal V}_b$ exhibits kinks which are tied to the domain evolution
present in the pure case, and also additional structure arising from the
formation of boson-fermion molecules, a prediction for future experiments.
