Function to me implies something with known characteristics. I suppose this is true when performing analytic FTs, but I think signal makes more sense when dealing with unknown inputs and numerical solutions. Regardless the relationship (or distinction) should be made clear when learning!
This is a good distinction to make. The term "signal" implies an information carrier (as in the expression "the signal in the noise"), which in light of information theory can be interpreted to mean an unknown variation (as modelled by, e.g. a stochastic process). This is philosophically distinct from the interpretation of a mathematical function as analytically defined and completely known.
