Chapter Review

We started this chapter with the big question:

What's a Function?

We have now answered that question in many different ways. In particular, we have seen that “function” is linguistically distinct from “formula.” Formulas often define functions, but not always. In fact, functions may be defined by

• formulas
• graphs
• data tables
• verbal descriptions
• conceptual relationships
• physical, biological, chemical, and other relationships

Functions are everywhere, and the concept of function is a powerful tool for dealing with the complexity we see all around us.

We learned in this chapter that all the following may be modeled or represented by formulas:

• relations between real-world variables that definitely are not functions
• relations that may not be functions
• relations that definitely are functions but have no apparent formulas

We can classify our functions by a trichotomy: real, observed, model. These functions have the characteristics summarized in Table 1.

We can classify our variables by a dichotomy: discrete or continuous. Some varying quantities jump from one value to the next, like a digital clock. These variables are discrete. They may have only a finite number of distinct values, or they may have infinitely many. Other varying quantities change smoothly, in such a way that there is never a next or a previous value. Such variables are called continuous. For example, we usually think of time proceeding in this way, and we model this change by the second hand on an analog clock.

The two clock types highlight the fact that we often model a single varying quantity (time, in this case) by both discrete and continuous variables. Our ability to move easily between discrete and continuous models will give us a broad range of conceptual and computational tools for representing and attacking the problems posed in this course.

Perhaps the most important — but still unstated — message of this chapter is that all of this effort is for something. We are going to deal with — and solve — problems that you will recognize as being important in a variety of different ways. In the process, we will see the power of mathematical abstraction to isolate essential features of a problem, to clear away irrelevant clutter, to lead to recognition of or reduction to a problem whose solution is already known. Our first step in that direction has been to come to grips with the simple, but extremely powerful, concept of FUNCTION.