Reviewing Domain and Range
Chapter 4 provided a discussion of how domain and range values can be understood in the context of relations. When you can establish a relation between a set you designate as a domain and another set you designate as a range, then you can usually create a function that defines the relation. A function constitutes a formalized relation between the values in the domain and the values in the range. As Figure 6.1 illustrates, you can depict this formalization using set notation. The expression f (x) reads, “the function of x” or, more briefly, “f of x.” It formally designates an equation that relates the domain and range. When you use such a function, you employ the values of the domain to generate the values of the range.
Figure 6.1. Set notation allows you to formalize how you express relations.
As discussed in Chapter 4, you identify an equation as a function. Here is an example of how to accomplish this:
f(x) = 3x + 2
You identify f(x) with the value the expression 3x = 2 generates. You might just as well write the function as y = 3x + 2. Expressed as a function, (f(x)) generates the value of y. It remains, however, that y is a variable that stands for the result of the application of f(x).
When you define a function, you can designate or describe its permissible domain and range values. For example, a function can serve as a way to relate an element from the set of real numbers to another element in the set of real numbers. With f(x) = 3x + 2, you can arrive at the following generalizations:
Domain = {All real numbers}
Range = {All real numbers}
Along narrower but equally formal lines, for an equation such as 
you can qualify the value of y so that it must be 0 or greater:
Domain = {All real numbers}
Range = {y | y ≥ 0}
The goal here is to designate that you cannot solve the equation for y if the value of x is less than 0.
Note
For much of the discussion in this book, consider function and equation to be nearly synonymous. Not all equations can be interpreted as functions, however. Generally, a function and an equation both relate the values of a range to the values of a domain.