Introduction

A functional equation is an equation in which the unknown objects are functions. Unlike differential equations, it is not a standard stand-alone topic in an undergraduate syllabus. Nonetheless, most students in mathematics would have encountered and solved some functional equations. For example, in an algebra course, we may have learnt how to obtain all the group homomorphisms from one group to another. A group homomorphism f is a function satisfying the algebraic identity f (xy) = f (x) f (y). When the mission is to find all such f, the identity is considered a functional equation and the unknown is f.

Each function comes with its domain and codomain. Variables are used to point to elements in the domain and codomain. In psychological sciences we are interested in people. Their psychological properties or qualities are the true variables. How the variables are measured, or how numbers are assigned to them, belongs to the theory of measurement. The numerically measured values are often, but not always, the variables of our functional equations. For this reason, in this chapter we treat functional equations with real variables more often. The function values may be a measure of some psychological attributes, and the codomain may then be the real numbers.

The real line, R, will be endowed with the usual algebraic, topological and ordering structures. This chapter does not deal with how and in what yardstick psychological properties are measured. If x represents how strong a stimulus is, we will skip the question whether 2x has the interpretation of “twice as strong.” Perhaps we should first address what is meant by “twice as strong” and ask if stimulus can be measured and scaled so that the interpretation holds. Tone intensity x and a respondent's perception of loudness, f (x), are measured along the psychological continuum. The articles by Hayes and Embretson (2012), Sowden (2012) and Van Zandt and Townsend (2012) offer excellent accounts on psychological measurement.

This chapter is intended to introduce and cover in depth some basic skills in handling functional equations. It is not a survey article. Several factors influenced our choices of the equations covered, the first being how basic and useful some methods are and whether they are appropriate at the senior undergraduate levels. The Cauchy and the Pexider equations are selected as an entry point.