Precalculus (6th Edition) Blitzer

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

Chapter 3 - Review Exercises - Page 512: 40

Answer

See the graph below:
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Work Step by Step

Draw the graph of $ f\left( x \right)=\ln x $ as follows: Construct the table of coordinates for $ f\left( x \right)={{\ln } }x $ and choose the appropriate values for $ x $ and calculate the corresponding $ y\text{-values}$. Now, plot every point provided in the above table and join them using a smooth curve, using the $ y\text{-axis}$ or $ x=0$ as the vertical asymptote. Now, the curve of $ g\left( x \right)=-\ln \left( 2x \right)$ in the graph above is obtained by horizontally shrinking the curve of $ f\left( x \right)=\ln x $ by a factor of 2 and then taking the reflection about the $ x\text{-axis}$. Divide the each x-coordinate by 2 units in the rectangular coordinate system. Multiply the y-coordinate by $-1$ and the curve $ g\left( x \right)=-\ln \left( 2x \right)$ is obtained as a reflection about the x-axis in the rectangular plane. Observing the above graph: The vertical asymptote is $ x=0$, domain is $\left( -\infty,0 \right)$, and range is $\left( -\infty,\infty \right)$ of the function $ f\left( x \right)=\ln x $ The vertical asymptote is $ x=0$, domain is $\left( -\infty,0 \right)$, and range is $\left( -\infty,\infty \right)$ of the function $ g\left( x \right)=-\ln \left( 2x \right)$.
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