Apoptosis or programmed cell death is an important process in multicellular organisms since it is involved in the decision to continue living or to commit suicide. It is an intrinsic part of cellular development and homeostasis, and failure in this mechanism can lead to serious disorders such as cancers, autoimmune diseases, and neurodegenerative disorders. The apoptotic pathway exhibits bistability, which is the capacity of a system to operate in two qualitatively distinct states. We analyze four basic mathematical models of apoptosis to study the processes that enable the system to perform the switch from survival to death. Specifically, we study how cooperativity and inhibition, which are key features of the apoptotic signal transduction network, enable the system to achieve bistability. Our contribution is the use of a purely analytical method to compute the steady states and eigenvalues of the Jacobian matrix at steady state, and thus we are able to completely determine regions in the parameter space where each of the considered models could exhibit bistability. We also discuss how changes in system parameters such as production and degradation rates affect the capacity of the system to exhibit bistability.