Special Function Forms

Not every function comes with one simple formula. Five special forms show up again and again throughout analysis and its applications, and this chapter introduces each of them: the composite function $f(g(x))$, which describes one dependency standing on top of another; the inverse function, which turns “the independent variable determines the dependent one” around; the piecewise function, which glues different expressions together on different intervals; the implicit function, which hides the dependent variable inside an equation; and the parametric form, which describes a curve by a pair of coordinates that both move with a parameter.

What unites them is a single question. The basic convention of a function is that one input gives one output, and each of these five forms relaxes that convention in a different direction while keeping the idea usable. So the chapter does two things for each form: it fixes the definition and the properties you can rely on, and it gives the test you use to decide which kind you are looking at. By the end you should be able to recognise which of the five a given expression belongs to, and reach for the right tool to work with it.