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 3.4 - Page 387: 23

Answer

Vertical asymptotes: $x = 0$ and $x = -4$ Graph holes: none

Work Step by Step

Vertical asymptotes in rational functions are always given by the denominator's zeros. Therefore, for this exercise, to identify the vertical asymptote, one equals each of the denominator's factors to zero: $$x(x + 4) = 0$$ And solve for each $x$: $$x = 0 ; (x + 4) = 0$$ $$x = 0 ; x = 0 - 4$$ $$ x = 0 ; x = -4$$ The "holes" in graphs, however, occur when a rational function behaves as a polynomial because the "numerator" can be divided into the "denominator" without a remainder. That is not the case in this exercise - therefore, there are no "holes".
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