Chapter 5: Further Topics in Functions

Before we embark upon any further adventures with functions, we need to take some time to gather our thoughts and gain some perspective. Chapter first introduced us to functions in Section. At that time, functions were specific kinds of relations - sets of points in the plane which passed the Vertical Line Test, Theorem. In Section, we developed the idea that functions are processes - rules which match inputs to outputs - and this gave rise to the concepts of domain and range. We spoke about how functions could be combined in Section using the four basic arithmetic operations, took a more detailed look at their graphs in Section and studied how their graphs behaved under certain classes of transformations in Section. In Chapter, we took a closer look at three families of functions: linear functions (Section ), absolute value functions (Section ), and quadratic functions (Section ). Linear and quadratic functions were special cases of polynomial functions, which we studied in generality in Chapter. Chapter culminated with the Real Factorization Theorem, Theorem, which says that all polynomial functions with real coefficients can be thought of as products of linear and quadratic functions. Our next step was to enlarge our field of study to rational functions in Chapter. Being quotients of polynomials, we can ultimately view this family of functions as being built up of linear and quadratic functions as well. So in some sense, Chapters,, and can be thought of as an exhaustive study of linear and quadratic functions and their arithmetic combinations as described in Section. We now wish to study other algebraic functions, such as $f(x)=\sqrt{x}$ and $g(x)=x^{2/3}$, and the purpose of the first two sections of this chapter is to see how these kinds of functions arise from polynomial and rational functions. To that end, we first study a new way to combine functions as defined below.