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Function diversity, or the range of tasks that individuals perform, is essential for productive organizations. In the absence of overarching principles, the characteristics of function diversity are seemingly unique to each domain. Here, we introduce an empirical framework and a mathematical model for the diversification of functions in a wide range of systems, such as bacteria, federal agencies, universities, corporations, and cities. Our findings reveal that the number of functions within these entities grows sublinearly with system size, with exponents ranging from 0.35 to 0.57, confirming Heaps' Law. In contrast, cities exhibit logarithmic growth in the occupation types. We generalize the Yule-Simon model to quantify a wide range of these empirical observations by introducing two new key attributes: a diversification parameter that characterizes the tendency for more populated functions to inhibit new function creation, and a specialization parameter that describes how a function's attractiveness depends on its abundance. These parameters allow us to position diverse systems, from microorganisms to metropolitan areas, within a two-dimensional abstract space. This mapping suggests underlying commonalities and differences in the foundational mechanisms that drive the growth of these systems.