In this work the polarizability of a subwavelength core-shell sphere is considered, where the shell exhibits a radially inhomogeneous permittivity profile. The mathematical treatment of the electrostatic polarizability is formulated in terms of the scattering potentials and the corresponding scattering amplitudes. As a result, a generalized expression of the polarizability is presented to be dependent of the radial inhomogeneity function. The extracted general model is applied for two particular cases, i.e., a power-law profile and a new class of permittivity profiles that exhibit exponential radial dependence. The proposed analysis quantifies in a simple manner the inhomogeneity effects, allowing the direct implementation of naturally or artificially occurring permittivity inhomogeneities for a wide range of applications within and beyond the metamaterial paradigm. Specifically, a special case of symmetric-antisymmetric resonant plasmonic degeneracy is identified and shown for the case of a core-shell sphere with a power-law permittivity profile. This degeneracy could be used for the experimental identification of inhomogeneity-induced effects or for applications where a strong coupling resonant regime is required. Furthermore, the described analysis opens avenues towards the phenomenological and first-principles modeling of the electrodynamic scattering effects for graded-index plasmonic particles at the nanoscale. Finally, such a description can be readily used either for the benchmarking of novel computational methods incorporating inhomogeneous materials or for inverse scattering purposes.