End Behavior and Local Behavior of Rational Functions

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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