A model representing the global average ionosphere and thermosphere (GAIT) is developed as a tool to explore the response of the coupled system to changes in the input solar irradiance between 3 and 360 nm. The GAIT model self-consistently solves the coupled continuity, momentum, and energy equations for the three major neutral species, N2, O2, and O, as well as minor neutral constituents important to the global energy budget. In the ionosphere the model includes five different ion species and two excited states of O+. The GAIT model also includes an approximate treatment of photoelectrons, in order to calculate secondary ionization and thermal electron volume heating rates. After examining the sensitivity of the GAIT model to uncertainties in key reaction rates and inputs, we consider its response to four different extreme ultraviolet irradiance models. With three of the irradiance models we reproduce the expected variation in exospheric temperature to within 2%; however, the dynamic range of the fourth is deemed to be too small. We conclude with an analysis of the model's wavelength-dependent sensitivity to input solar photons.