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 399: 72


Graph the function as:

Work Step by Step

Step 1: Substitute $x=-x.$ $\begin{align} & f\left( x \right)=\frac{x-4}{{{x}^{2}}-x-6} \\ & f\left( -x \right)=\frac{\left( -x \right)-4}{{{\left( -x \right)}^{2}}-\left( -x \right)-6} \\ & =\frac{-\left( x+4 \right)}{{{x}^{2}}+x-6} \end{align}$ and $\begin{align} & -f\left( x \right)=-\left( \frac{x-4}{{{x}^{2}}-x-6} \right) \\ & \ne f\left( -x \right) \end{align}$ Hence, the graph of the function is symmetric about neither $y\text{-}$ axis nor origin. Step 2: To calculate the x intercepts equate $f\left( x \right)=0$. $\begin{align} & \frac{x-4}{{{x}^{2}}-x-6}=0 \\ & x=4 \end{align}$ Step 3: To calculate the y intercepts evaluate $f\left( 0 \right)$. $f\left( 0 \right)=\frac{2}{3}$ Step 4: Since the degree of the numerator is less than the denominator, there is no horizontal asymptote. Step 5: For the vertical asymptote, equate the denominator to 0. $x=-2,3$
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