This paper aims at providing a simplified analytical solution for functionally graded beam stress analysis and optimized material gradation on the beam deflection. The power-law (P-FGM) and exponential (E-FGM) material functions were considered for an exact solution of the normal and shear stress distributions across the beam thickness. Optimization of material function on the FGM beam deflection, which is new of its kind, was also investigated considering both simply supported and cantilever beams. It was observed that the non-dimensional normal stress and shear stress are independent of the elastic moduli values of the constituent materials but rather depends on both the ratio of the elastic moduli and the location across the beam thickness in the E-FGM material function model. This observation was first validated from available kinds of literature and through numerical simulation using ABAQUS and extended to the P-FGM stress analysis. The maximum deflection on the FGM beam occurred for a homogenous steel beam while the minimum deflection was observed on the beam with a P-FGM material function. The results of this work demonstrate that if properly designed and optimized, FGMs can provide an alternative material solution in structural applications.