Precalculus (6th Edition) Blitzer

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

Chapter 6 - Section 6.7 - The Dot Product - Exercise Set - Page 794: 89


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Work Step by Step

The function is, $f\left( x \right)=\frac{4x-4}{x-2}$, Replace x as $-x$ to check the symmetry. $\begin{align} & f\left( -x \right)=\frac{4\left( -x \right)-4}{-x-2} \\ & =\frac{-\left( 4x+4 \right)}{-\left( x+2 \right)} \\ & =\frac{4x+4}{x+2} \end{align}$ Since $f\left( x \right)\ne f\left( -x \right)$, therefore graph is not in symmetry to the y-axis. Now put $x=0$ and get the y-intercept. $\begin{align} & f\left( x \right)=\frac{4x-4}{x-2} \\ & =\frac{4\times 0-4}{0-2} \\ & =\frac{-4}{-2} \\ & =2 \end{align}$ Thus, the graph of the function passes through the point $\left( 0,2 \right)$. Substitute $0$ for $f\left( x \right)$ to get $x-\text{intercept}$. $\begin{align} & 4x-4=0 \\ & 4x=4 \\ & x=1 \end{align}$ Hence the graph passes through the point $\left( 1,0 \right)$ For the vertical asymptotes the denominator is equal to zero, so to find out the vertical asymptotes, $\begin{align} & x-2=0 \\ & x=2 \end{align}$ The vertical asymptote $x=2$. Take the ratio of the coefficient of x in the numerator and denominator to get the horizontal asymptotes. $\begin{align} & y=\frac{4}{1} \\ & =4 \end{align}$ Collect all the points of the vertical asymptotes(x) and horizontal asymptotes(y).
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