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Find all the vertical and horizontal asymptotes of the function f(x) = \frac{x^2 - 4}{3x^2 + 3x...

Question:

Find all the vertical and horizontal asymptotes of the function {eq}f(x) = \frac{x^2 - 4}{3x^2 + 3x - 6} {/eq}

Obtaining the Asymptotes of a Rational Function:

The behavior of rational functions are sometimes distinguished from their asymptotes. Asymptotes are lines that are very close to a curve yet does not touch it. They are common in rational functions due to the restrictions in the domain of the function. Vertical asymptotes are achieved by equating the denominator to zero. Meanwhile, horizontal asymptotes are present when the highest exponent in the numerator is the same as the denominator. Lastly, oblique asymptotes will be observed if the highest exponent in the numerator is greater than the highest exponent in the denominator. It is most important that the asymptotes are obtained from the simplest form of the rational expression.

Answer and Explanation:

Before obtaining the asymptotes, we first simply the rational function by factoring the numerator and denominator. So,

{eq}\begin{align} f(x) &=...

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Analyzing the Graph of a Rational Function: Asymptotes, Domain, and Range

from Math 105: Precalculus Algebra

Chapter 4 / Lesson 9
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