Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 3 - Section 3.6 - Rational Expressions - 3.6 Exercises - Page 309: 50

Answer

x-intercept: NONE y-intercept: (0, -1) Vertical Asymptote: x=6, -1 Domain: (-∞, -1) U (-1, 6) U (6, ∞) Horizontal Asymptote: y=0 Range: (-∞, 0) U (0, ∞) See graph below

Work Step by Step

$r(x)=\frac{6}{x^2 - 5x - 6}$ No x-intercept, numerator acan never be equal to zero y-intercept is the ratio of the constants, which is 6 / -6 = -1 Thus, the y-intercept is at (0, -1) Vertical asymptotes are when the denominator is equal to 0 $(x^2 - 5x - 6) = 0$ $(x-6)(x+1) = 0$ x = 6, -1 So, domain is from (-∞, -1) U (-1, 6) U (6, ∞) Horizontal asymptote is the ratio of the constants of the leading term (with equal degree) 0 (since top degree is not 2, unlike bottom) Thus, the horizontal asymptote is at y=0 So the range is from (-∞, 0) U (0, ∞)
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