Chapter 5: Further Topics on Functions

Up until this point in the text, we have primarily focused on studying particular families of functions. These families and their relationships to one another provide useful examples of more abstract function structures and relationships. The notions introduced in this chapter will not only provide us a more formal vocabulary with which to describe the connections between the function families we have already studied, but, more importantly, give us additional lenses through which to view new families of functions that we'll encounter.

In this section, we review of the concepts associated with the graphs of functions. We introduced the notion of the graph of a function in Section 1.1, and the vast majority of the graphs we have encountered in this text were generated from an algebraic representation of a function. In this section, we define the functions geometrically from the outset and review the important concepts associated with the graphs of functions.

Recall the domain of a function is the set of inputs to the function and the range of a function is the set of outputs from the function. When graphing a function whose domain and range are subsets of real numbers, we plot the ordered pairs $(\text{input},\text{output})$ on the Cartesian plane. Hence, the domain values are found on the horizontal axis while the range values are found on the vertical axis.

Recall from Definition 1.3 that the largest output from the function (if there is one) is called the maximum or, when there may be some confusion, the absolute maximum of the function. Likewise, the smallest output from the function (again, if there is one) is called the minimum or absolute minimum.