End Behavior and Local Behavior of Rational Functions

In the first section on rational functions, you will learn about their general characteristics and how to use standard notation to describe them. When you are finished, you will be able to use arrow notation to describe long-run behavior given a graph or an equation. You will also be able to use standard notation to describe local behavior.

Using Arrow Notation

We have seen the graphs of the basic reciprocal function and the squared reciprocal function from our study of toolkit functions. Examine these graphs, as shown in Figure 1, and notice some of their features.

Figure 1

Several things are apparent if we examine the graph of $f(x)=\frac{1}{x}$ .

1. On the left branch of the graph, the curve approaches the $x$ -axis $(y=0)$ as $x \rightarrow-\infty$ .

2. As the graph approaches $x=0$ from the left, the curve drops, but as we approach zero from the right, the curve rises.

3. Finally, on the right branch of the graph, the curves approaches the $x$ -axis $(y=0)$ as $x \rightarrow \infty$ .

To summarize, we use arrow notation to show that $x$ or $f(x)$ is approaching a particular value. See Table 1.

Symbol	Meaning
$x \rightarrow a^{-}$	$x$ approaches $a$ from the left ( $x < a$ but close to $a$ )
$x \rightarrow a^{+}$	$x$ approaches $a$ from the right ( $x > a$ but close to $a$ )
$x \rightarrow \infty$	$x$ approaches infinity ( $x$ increases without bound)
$x \rightarrow-\infty$	$x$ approaches negative infinity ( $x$ decreases without bound)
$f(x) \rightarrow \infty$	the output approaches infinity (the output increases without bound)
$f(x) \rightarrow-\infty$	the output approaches negative infinity (the output decreases without bound)
$f(x) \rightarrow a$	the output approaches $a$

Table 1