Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter 7 - Trigonometric Identities and Equations - 7.5 Inverse Circular Functions - 7.5 Exercises: 9

Answer

a. $(-\infty, \infty) $ b. $(-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2})$ c. increasing d. no

Work Step by Step

See figure $20$ on p.$702.$ (or the table on page 703 ) In order to have an inverse, the domain of $\tan x$ is restricted to .$(-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2})$. $y=\tan^{-1} x$ ($y$ is the number from $(-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2})$ for which $\tan y=x$) (a) and (b) Domain: $(-\infty, \infty) $ Range:$ (-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2})$ Quadrants (unit circle): I and IV (c) Figure $20$: $\tan^{-1}x$ is increasing. For part (d), see the domain. All real numbers are in the domain. There is no x for which $\tan^{-1}x$ is not defined.
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