Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 2 - Section 2.6 - Rational Functions and Their Graphs - Exercise Set - Page 400: 93


$\frac{x-2}{x+2}$; see graph

Work Step by Step

Step 1. Multiply the numerator and the denominator with $(x+2)(x-2)$. We have $f(x)=\frac{x^2-4-3(x-2)}{x^2-4+x+2}=\frac{x^2-3x+2}{x^2+x-2}=\frac{(x-1)(x-2)}{(x+2)(x-1)}=\frac{x-2}{x+2}$ where $x\ne 1, \pm2,$ with holes at $(1, -\frac{1}{3})$ and $(2,0)$ Step 2. We can identify a vertical asymptote as $x=-2$ Step 3. We can also identify a horizontal asymptote as $y=1$ Step 4. We can find the y-intercept at $y=-1$ Step 5. Testing signs across the vertical asymptotes, we have $...(+)...(-2)...(-)...$ Step 6. Based on the above results, we can graph the function as shown in the figure.
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