Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 1 - Functions - 1.4 Trigonometric Functions and Their Inverse - 1.4 Exercises - Page 48: 13

Answer

$(x\rightarrow \pm\infty)\Rightarrow \left(f^{-1}(x)\rightarrow \pm\dfrac{\pi}{2}\right)$

Work Step by Step

We are given the function: $f(x)=\tan x$ For the inverse to exist, we must restrict the domain of $f$ so that the function is one-to-one: $D_f=\left(\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$ $R_f=(-\infty,\infty)$ The domain and range of the inverse $f^{-1}(x)=\tan^{-1}(x)$ are: $D_{f^{-1}}=(-\infty,\infty)$ $R_{f^{-1}}=\left(\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$ The graph of $f^{-1}$ is symmetrical with the graph of $f$ over the line $y=x$. When $x$ gets close to $\pm\dfrac{\pi}{2}$, $f$ is very large or very small. When $x$ is very large or very small, $f^{-1}$ gets close to $\pm\dfrac{\pi}{2}$.
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