The need to model and to reason about design alternatives throughout the design process demands robust representation schemes of function, behavior, and structure. Function describes the physical effect imposed on an energy or material flow by a design entity without regard for the working principles or physical solutions used to accomplish this effect. Behaviors are the physical events associated with a physical artifact (or hypothesized concept) over time (or simulated time) as perceived by an observer. Structure, the most tangible concept, partitions an artifact into meaningful constituents such as features, Wirk elements, and interfaces in addition to the widely used assemblies and components. The focus of this work is on defining a model for function-based representations that can be used across various design methodologies and for a variety of design tasks throughout all stages of the design process. In particular, the mapping between function and structure is explored and, to a lesser extent, its impact on behavior is noted. Clearly, the issues of a function-based representation's composition and mappings directly impact certain computational synthesis methods that rely on (digitally) archived product design knowledge. Moreover, functions have already been related to not only form, but also information of user actions, performance parameters in the form of equations, and failure mode data. It is essential to understand the composition and mappings of functions and their relation to design activities because this information is part of the foundation for function-based methods, and consequently dictates the performance of those methods. Toward this end, the important findings of this work include a formalism for two aspects of function-based representations (composition and mappings), the supported design activities of the model for function-based representations, and examples of how computational design methods benefit from this formalism.