College Algebra (6th Edition)

Published by Pearson
ISBN 10: 0-32178-228-3
ISBN 13: 978-0-32178-228-1

Chapter 3 - Polynomial and Rational Functions - Exercise Set 3.5 - Page 406: 24


Vertical asymptotes$:\quad x=0$ and $x=3$ no holes

Work Step by Step

Locating Vertical Asymptotes, (page 395) tells us that for $f(x)=\displaystyle \frac{p(x)}{q(x)}$, if p(x) and q(x) have NO common factors, and a is a zero of the denominator $q(x)$ , then the line $x=a$ is a vertical asymptote. BUT, if they DO have common factors, after REDUCING the form of the function's equation, the number a may not cause the denominator to be zero any more. in which case there will be a hole at x=a. The point is that both p(x) and q(x) have to be factored, and if we can, then we reduce the expression for f(x). ------------------ $g(x)=\displaystyle \frac{x+3}{x(x-3)}$ $p(x)=x+3\qquad $... fully factored $q(x)=x(x-3)\qquad $... fully factored No common factors, no holes. $q(x)=0$ $x(x-3)=0$ $x=0$ or $x=3$ Vertical asymptotes$:\quad x=0$ and $x=3$
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