A class of real functions, which is the generalization of a family of convex functions, is introduced; in this connection, we have defined X-convex, strictly X-convex, quasi-Xconvex, strictly quasi-X-convex, and semi-strictly quasi-X-convex functions. Moreover, in this paper, we give a detailed study of the fundamental properties of these functions with various examples, supporting the concepts. Finally, the study of optimization problems employs quasi-X-convex, semistrictly quasi-X-convex, and strictly quasi-X-convex functions.