Mathematical theories are essential for explanations in physics, chemistry andengineering. These theories often incorporate functions that are defined by theirrelation to other variables in the theory but not with reference to experimentalobservations. The wave function in quantum mechanics is perhaps one of thebest known example of such function, even though classical theories also providemany examples of such functions. These functions, which seem to hang in thin airdisconnected to experimental data, offer a daunting challenge to the instructor. Inthis article we consider the epistemic status of such functions and a method ofintroducing them to the students, a method that does not distort the original theory.We build our model for explanation on Hempel's analysis of relation between theoryand experiment and refine it further to show their roles in concept formation.