The residual stresses induced in functionally graded medium (FGM) with inhomogeneity cooling down from the processing temperature are determined with concentric cylinder model and analytical solutions of the inhomogeneous governing equations for displacement components. The analytical solutions derived here are general for power-law variations of the elastic moduli of the FGM. With a power exponent, analytical expressions for the residual stresses of FGM with inhomogeneity can be obtained. By changing the power exponent and the coefficient of the power terms, the solutions obtained here could be applied to different properties of FGM with inhomogeneity. The results show that the huge difference exists between FGM with inhomogeneity and homogeneous medium with inhomogeneity. The variations of FGM and inhomogeneity size have a great deal of effect on the residual stresses in FGM.