  We investigate the radio-frequency spectroscopy of impurities interacting
with a quantum gas at finite temperature. In the limit of a single impurity, we
show using Fermi's golden rule that introducing (or injecting) an impurity into
the medium is equivalent to ejecting an impurity that is initially interacting
with the medium, since the "injection" and "ejection" spectral responses are
simply related to each other by an exponential function of frequency. Thus, the
full spectral information for the quantum impurity is contained in the
injection spectral response, which can be determined using a range of
theoretical methods, including variational approaches. We use this property to
compute the finite-temperature equation of state and Tan contact of the Fermi
polaron. Our results for the contact of a mobile impurity are in excellent
agreement with recent experiments and we find that the finite-temperature
behavior is qualitatively different compared to the case of infinite impurity
mass.
