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: 31


$\left.\begin{array}{ccc} & domain & range\\ \hline f(x) & (-\infty,\infty) & (0,\infty)\\ g(x) & (0,\infty) & (-\infty,\infty) \end{array}\right.$

Work Step by Step

$ f(x)=a^{x}$ and $ g(x)=\log_{a}x $ are inverse functions. Graphs of inverse functions are reflections of each other, over the line $ y=x.$ The graph of $(\displaystyle \frac{1}{3})^{x}$ is always above the x-axis, falls from the far left rapidly, passes through $(-3,27),(-2,9),(-1,3)$, intersects the y-axis at $(0,1)$, continues to fall toward the x-axis (never touching it), etc. Plan: Plot some points $(x,(\displaystyle \frac{1}{3})^{x})$ and join with a smooth curve to graph $ f(x).$ Graph the line $ y=x.$ Reflect the points plotted earlier about the line $ y=x.$ Join these new points to graph $ g(x)=\log_{1/3}x $ From the graph, read the domains and ranges, $\left.\begin{array}{lll} & domain & range\\ \hline f(x) & (-\infty,\infty) & (0,\infty)\\ g(x) & (0,\infty) & (-\infty,\infty) \end{array}\right.$
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