Section 3: Graphing Rational Functions

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 7.3 Graphing Rational Functions 639 Version: Fall 2007 7.3 Graphing Rational Functions We’ve seen that the denominator of a rational function is never allowed to equal zero; division by zero is not defined. So, with rational functions, there are special values of the independent variable that are of particular importance. Now, it comes as no surprise that near values that make the denominator zero, rational functions exhibit special behavior, but here, we will also see that values that make the numerator zero sometimes create additional special behavior in rational functions. We begin our discussion by focusing on the domain of a rational function. The Domain of a Rational Function When presented with a rational function of the form f ( x ) = a 0 + a 1 x + a 2 x 2 + · · · + a n x n b 0 + b 1 x + b 2 x 2 + · · · + b m x m , (1) the first thing we must do is identify the domain. Equivalently, we must identify the restrictions , values of the independent variable (usually x ) that are not in the domain. To facilitate the search for restrictions, we should factor the denominator of the rational function (it won’t hurt to factor the numerator at this time as well, as we will soon see). Once the domain is established and the restrictions are identified, here are the pertinent facts. Behavior of a Rational Function at Its Restrictions . A rational function can only exhibit one of two behaviors at a restriction (a value of the independent variable that is not in the domain of the rational function). 1. The graph of the rational function will have a vertical asymptote at the re- stricted value. 2. The graph will exhibit a “hole” at the restricted value. In the next two examples, we will examine each of these behaviors. In this first example, we see a restriction that leads to a vertical asymptote. l⚏ Example 2. Sketch the graph of f ( x ) = 1 x + 2 . The first step is to identify the domain. Note that x = 2 makes the denominator of f ( x ) = 1 / ( x + 2) equal to zero. Division by zero is undefined. Hence, x = 2 is not in the domain of f ; that is, x = 2 is a restriction. Equivalently, the domain of f is { x : x = 2 } . Now that we’ve identified the restriction, we can use the theory of Section 7.1 to shift the graph of y = 1 /x two units to the left to create the graph of f ( x ) = 1 / ( x + 2) , as shown in Figure 1 . Copyrighted material. See: 1

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640 Chapter 7 Rational Functions Version: Fall 2007 x 5 y 5 x = 2 y =0 Figure 1. The function f ( x ) = 1 / ( x + 2) has a restriction at x = 2 . The graph of f has a vertical asymptote with equation x = 2 . The function f ( x ) = 1 / ( x + 2) has a restriction at x = 2 and the graph of f exhibits a vertical asymptote having equation x = 2 . It is important to note that although the restricted value x = 2 makes the de- nominator of f ( x ) = 1 / ( x + 2) equal to zero, it does not make the numerator equal to zero. We’ll soon have more to say about this observation.
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