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: 56

Answer

x-intercept: None y-intercept: (0, -2) Vertical Asymptote: x=-1, 3 Domain: (-∞, -1) U (-1, 3) U (3, ∞) Horizontal Asymptote: y=3 Range: (-∞, 3) U (3, ∞) See graph below

Work Step by Step

$r(x)=\frac{3x^2 + 6}{(x^2 -2x - 3)}$ $3x^2 + 6 = 0$ $3(x^2 + 2) = 0$ No x-intercept y-intercept is the ratio of the constants, which is 6/-3 = -2 Thus, the y-intercept is at (0, -2) Vertical asymptotes are when the denominator is equal to 0 $x^2 -2x - 3 = 0$ $(x-3)(x+1) = 0$ x = 3, -1 So, domain is from (-∞, -1) U (-1, 3) U (3, ∞) Horizontal asymptote is the ratio of the constants of the leading term (with equal degree) 3/1 = 3 Thus, the horizontal asymptote is at y=3 So the range is from (-∞, 3) U (3, ∞)
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