We present analytic expressions for the frequency-domain transfer function of semiconductor optical amplifier Mach-Zehnder interferometric (SOA-MZI) switches that employ a single optical control signal and a continuous wave input optical beam. Our analysis relies on first-order perturbation theory approximations applied both to the SOA response as well as to the SOA-MZI characteristics, yielding a frequency response that enables a qualitative insight into the different SOA-MZI operational regimes. The final transfer function expression is utilized for the analysis and evaluation of the multifunctional potential of SOA-MZI switches, concluding with the necessary conditions for supporting a number of completely different SOA-MZI-based nonlinear signal processing applications that have been demonstrated experimentally: wavelength conversion, packet envelope detection (PED), and clock recovery (CR). The theoretically obtained operational conditions are in close agreement with experimental observations, showing that SOA-MZIs can serve as functional circuit elements in applications with different requirements depending on its operational parameters: as low-pass filtering devices with cut-off frequencies in the megahertz regime or in the multi-gigahertz regime, and as resonant modules resembling band-pass filtering structures. The validity of our theoretical SOA-MZI frequency-domain system model is further confirmed by its successful incorporation in a Fabry-Perot assisted SOA-MZI subsystem, demonstrating PED and CR operations through the exploitation of typical systems theory tools.