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: 26

Answer

Vertical asymptote: $x=3$ Hole at $x=0$

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). ------------------ $h(x)=\displaystyle \frac{x}{x(x-3)}$ $p(x)=x\qquad $... fully factored $q(x)=x(x-3)\qquad $... fully factored We can reduce the expression for h(x): $h(x)=\displaystyle \frac{x}{x(x-3)}=\frac{1}{x-3}, \quad x\neq 0$ the denominator is zero for $x=3,$ h(x) is undefined for $x=0$ Vertical asymptote: $x=3$ Hole at $x=0$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.