This study systemically presents an inverse homogenization method in the design of functional gradient materials, which gained substantial attention recently due to their layer-by-layer defined physical properties. Each layer of these materials is unilaterally constructed by periodically extended microstructural elements (namely base cells), whose effective properties can be decided by the homogenization theory in accordance with the material distribution within the base cell. The design objective is to minimize the summation of the least squares of the difference between corresponded entries in target and effective elasticity tensors. The method of moving asymptote drives the minimization of this positive objective function, which forces the effective values approach to the targets as closely as possible. The sensitivity of the effective elasticity tensors with respect to the design variables is derived from the adjoint variable method and it guides the minimization algorithm efficiently. To guarantee the connectivity between adjacent layers, non-design domains occupied by solid materials acting as connective bars are fixed in the design of base cells. Furthermore, nonlinear diffusion technique is introduced to avoid checkerboard patterns and blur boundaries in the microstructures. A series of two-dimensional examples targeted for the elasticity tensors with same extreme Poisson ratios but different densities in each layer are illustrated to highlight the computational material design procedure.