The notion of a function is central to modern mathematics and the usual modern definition involves a rule which takes each element of the domain X and assigns a unique element in the range Y. Examples where X and Y are both the set of real numbers are f(x)=x3 or f(x)=ex. Just over one hundred years ago G. H. Hardy [1] made the following remarks about functions. We must point out that the simple examples of functions mentioned above possess three characteristics which are by no means involved in the general idea of a function viz:
- y is determined for every value of x;
- to each value of x for which y is given corresponds one and only one value of y;
- the relation between x and y is expressed by means of an analytical formula.
[...] All that is essential is that there should be some relation between x and y such that to some values of x at any rate correspond values of y.
When I read this I was rather surprised that Hardy took a somewhat more liberal view of the idea of a function than we commonly do today. He gives a number of further examples some of which involve formulae an equation or algebraic expression in x and y including an infinite series. Others involve a relationship between x and y which follow from some geometrical construction. In this article I’d like to give some examples of geometrical functions as illustrated by the software GeoGebra.
