Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 6 - Inverse Circular Functions and Trigonometric Equations - Section 6.1 Inverse Circular Functions - 6.1 Exercises - Page 265: 72

Answer

For the function $csc^{-1}~2x$: The domain is $(-\infty, -0.5]\cup[0.5,\infty)$ The range is $[-\frac{\pi}{2},0)\cup (0, -\frac{\pi}{2}]$ We can see a sketch of the graph of $~~csc^{-1}~2x~~$ below. Note there is a horizontal asymptote at $y = 0$
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Work Step by Step

Consider the function $csc~2x$: The domain is all real numbers except $~~\frac{\pi}{2}~n~~$, where $n$ is an integer The range is $(-\infty, -1]\cup[1,\infty)$ We can consider the function $~~csc^{-1}~2x~~$ with the domain restricted to $[-\frac{\pi}{4},0)\cup (0, -\frac{\pi}{4}]$ Then for the function $csc^{-1}~2x$: The domain is $(-\infty, -0.5]\cup[0.5,\infty)$ The range is $[-\frac{\pi}{2},0)\cup (0, -\frac{\pi}{2}]$ We can see a sketch of the graph of $~~csc^{-1}~2x~~$ below. Note there is a horizontal asymptote at $y = 0$
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