All products and artifacts are designed for a purpose. There is some intended reason behind their existence: the product or artifact function. Functional modeling provides an abstract, yet direct, method for understanding and representing an overall product or artifact function. Function modeling also provides a strategy for problem decomposition, physical modeling, product architecting, concept generation, and team organization. A formal function representation is needed to support function modeling, and a standardized set function-related terminology is necessary to achieve repeatable and meaningful results from such a representation. We refer to this representation as a functional basis; in this paper, we seek to reconcile and integrate two independent research efforts into a significantly evolved functional basis. These efforts include research from the National Institute of Standards and Technology (NIST) and two U.S. universities, and their industrial partners. The overall approach for integrating the functional representations is developed, in addition to the final results. The integration process is discussed relative to differences, similarities, insights into the representations, and product validation. Based on the results, a more versatile and comprehensive design vocabulary is obtained. This vocabulary will greatly enhance and expand the frontiers of research in design repositories, product architecture, design synthesis, and general product modeling.