Technical Mathematics, Sixth Edition

Chapter 4. Functions

OBJECTIVES When you have completed this chapter, you should be able to Distinguish between relations and functions. Recognize different forms of a function: equation, table of point pairs, verbal statement, graph. Use functional notation. Distinguish between implicit and explicit forms of an equation. Rewrite a simple implicit equation in explicit form. Substitute into a function. Manipulate a function. Manipulate functions by calculator. Write a composite function. Find the inverse of a function. Find the domain and range of a function.

The equations we have been solving in the last few chapters have contained only one variable. For example,

x(x − 3) = x² + 7

contains the single variable x.

But many situations involve two (or more) variables that are somehow related to each other. Look, for example, at Fig. 4-1, and suppose that you leave your campsite and walk a path up the hill. As you walk, both the horizontal distance x from your camp, and the vertical distance y above your camp, will change. You cannot change x without changing y, and vice versa (unless you jump into the air or dig a hole). The variables x and y are related.

In this chapter we study the relation between two variables and introduce the concept of a function. The idea of a function provides us with a different way of speaking about mathematical relationships. We could say, for example, that the formula for the area of a circle as a function of its radius is A = πr².

Figure 4.1. FIGURE 4-1