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Astrophysical simulations of convection frequently impose different thermal boundary conditions at the top and the bottom of the domain in an effort to more accurately model natural systems. In this work, we study Rayleigh-Benard convection (RBC) under the Boussinesq approximation. We examine simulations with mixed temperature boundary conditions in which the flux is fixed at the bottom boundary and the temperature is fixed at the top ("FT"). We aim to understand how FT boundaries change the nature of the convective solution compared to the traditional choice of thermal boundaries, in which the temperature is fixed at the top and bottom of the domain ("TT"). We demonstrate that the timescale of thermal relaxation for FT simulations is dependent upon the initial conditions. "Classic" initial conditions which employ a hydrostatically -- and thermally -- balanced linear temperature profile exhibit a long thermal relaxation. This long relaxation is not seen in FT simulations which use a TT simulation's nonlinear state as initial conditions ("TT-to-FT"). In the thermally relaxed, statistically stationary state, the mean behavior of an FT simulation corresponds to an equivalent simulation with TT boundaries, and time- and volume-averaged flow statistics like the Nusselt number and the Peclet number are indistinguishable between FT and TT simulations. FT boundaries are fundamentally asymmetric, and we examine the asymmetries that these boundaries produce in the flow. We find that the fixed-flux boundary produces more extreme temperature events than the fixed-temperature boundary. However, these near-boundary asymmetries do not measurably break the symmetry in the convective interior. We briefly explore rotating RBC to demonstrate that our findings with respect to thermal relaxation carry over to this more complex case, and to show the power of TT-to-FT initial conditions.