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.

Learning Objectives

In this section, you will:

Use arrow notation.
Solve applied problems involving rational functions.
Find the domains of rational functions.
Identify vertical asymptotes.
Identify horizontal asymptotes.
Graph rational functions.

Suppose we know that the cost of making a product is dependent on the number of items, $x$ , produced. This is given by the equation $C(x)=15,000 x-0.1 x^{2}+1000$ . If we want to know the average cost for producing $x$ items, we would divide the cost function by the number of items, $x$ .

The average cost function, which yields the average cost per item for $x$ items produced, is

$f(x)=\frac{15,000 x-0.1 x^{2}+1000}{x}$

Many other application problems require finding an average value in a similar way, giving us variables in the denominator. Written without a variable in the denominator, this function will contain a negative integer power.

In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.

Source: Rice University, https://openstax.org/books/college-algebra/pages/5-6-rational-functions
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