Introduction

Sets, relations and functions sit at the heart of modern mathematical thinking, but the idea of a function did not arrive fully formed. As the mathematician Luzin observed, the concept underwent profound changes over time.

• Galileo (1564-1642) studied how the position of a planet depends on time -- an early, physical use of "one quantity determined by another".
• Descartes (1596-1650) showed that an equation in two variables, drawn as a curve, expresses exactly this kind of dependence.
• Leibnitz (1646-1716) was the first to use the word "function" (in a 1673 manuscript), for any quantity varying along a curve.
• Dirichlet (1805-1859), a student of Gauss, gave the first "formal" definition using the now-familiar notation $y=f(x)$.
• In the 20th century this was widened further, to any arbitrary correspondence -- between numerical or non-numerical values -- that satisfies a uniqueness condition.
• Cantor (1845-1918) built set theory, after which mathematicians re-cast "correspondence" as "relation", and a function became a special kind of relation.

Before turning to relations and functions themselves, this chapter first recalls -- in more depth -- what was learned earlier about sets, Venn diagrams and Cartesian products. It then gives relations and functions a fresh, rigorous treatment: constants, variables, intervals and neighbourhoods; the various types of relations (reflexive, symmetric, transitive, equivalence) and how to construct a relation of a required type; the modern definition of a function through relations and its different representations; elementary and special functions, one-to-one/onto/bijective functions, composition and inverses, algebra of functions; and finally how to identify and sketch the graph of a complicated function using reflection, translation and dilation of a simpler, known graph.