The Basic Concepts of Functions

A function is the basic mathematical model for describing how one quantity depends on another, and it is also one of the core concepts of higher mathematics. This chapter settles four questions: what determines a function (its domain, its correspondence rule and its range), what ways of representing the same function there are (formula, graph, table and list), how those representations convert into one another, and why “two expressions that look different” may represent the same function.

Once these four things are straight, the later chapters on properties, transformations and the various kinds of specific functions will not be a matter of memorising conclusions without understanding why they hold. The aim of this chapter is that by the end of it you should be able to find the domain of a function yourself, work out its range, convert back and forth between the formula and the graph, and say clearly when two expressions represent the same function.