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 407: 32


Vertical asymptote: $x=-5$ Hole at $x=5$

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). ------------------ $g(x)=\displaystyle \frac{x-5}{x^{2}-25}$ $p(x)=x-5\qquad $... fully factored $q(x)=x^{2}-25=(x+5)(x-5)\qquad $ (difference of squares) We can reduce the expression for $g(x):$ $g(x)=\displaystyle \frac{x-5}{x^{2}-25}=\frac{(x-5)}{ (x+5)(x-5)}=\frac{1}{x+5},\quad x\neq 5$ Vertical asymptote: $x=-5$ Hole at $x=5$
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