Structures made of functionally graded materials have attracted much interest recently. The idea is to create a material that fulfills a specified function in accordance to the identified purpose of structure's utilization. In this study, a semi-inverse problem is posed of determining the needed variation of the axial grading of an inhomogeneous column subjected to a central force, generalizing the ungraded column case. Remarkably, it turns out that for a specific combination of parameters, there could exist three different axially graded columns, that possess the same buckling load. Whereas this fact is not immediately apparent, the proposed formulation leads to the design of the axially graded column in such a manner that the buckling load is not less than a prespecified load. Purpose-oriented design demands columns have at least a prespecified buckling load. To solve the problem, a combined analytical-numerical procedure is developed in this study.