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The aim of this study is to explore the possibility of introducing the general notions of function and inverse function through a mathematical activity on linear functions, focusing on the quantitative meaning associated to the connection between a relation and its inverse. We present a genetic decomposition, that is, a viable cognitive path for learning these introductory concepts, in terms of mental structures and mechanisms, from the viewpoint of APOS (Action—Process—Object—Schema) theory. One of the innovative aspects of this study consists in the design of a genetic decomposition that involves the construction of two concepts. Through interviews, we explore the conceptions that two high school students—who had not been introduced to the function concept before—developed in connection with the notions in question. Our results confirm that not only it is possible to study a relation and its inverse together, but also doing that can enhance the understanding of the meanings involved in the mathematical relation. This leads us to question the learning of mathematics being linear, and to open a discussion about the organization of mathematical content for instructional purposes. We offer suggestions for further research both involving other mathematical concepts, and in terms of theoretical constructs.