Question

Range of $f(x) = sin^{-1} x + tan^{-1} x + sec^{-1} x$ is

• A. $\left(\frac{\pi}{4}, \frac{3\pi }{4}\right)$
• B. $\left[\frac{\pi}{4}, \frac{3\pi }{4}\right]$
• C. $\left\{\frac{\pi}{4}, \frac{3\pi }{4}\right\}$
• D. None of these

Answer 1

The Correct Option is C

$f(x ) = sin^{-1}x + tan^{-1}x + sec^{-1}x$ Domain of $sin^{-1}x = [-1,1]$ Domain of $tan^{-1}x = ( -\infty, \infty)$ Domain of $sec^{-1} = ( -\infty, \infty) - (-1,1)$ Domain of $f(x) = [ - 1, 1] \cap (-\infty, \infty) \cap [(-\infty, \infty) - (-1, 1)]$ $ = \{-1,1\}$ Now $f(-1 ) = sin^{-1} + tan^{-1}( - 1 ) + sec^{-1}( -1 )$ $= -\frac{\pi}{2} - \frac{\pi }{4} +\pi = \frac{\pi }{4}$ and $f\left(1\right) = sin^{-1}\left(1\right) + tan^{-1}\left(1\right) + sec^{-1}\left(1\right)$ $= \frac{\pi }{2} + \frac{\pi }{4} + 0$ $= \frac{3\pi }{4}$ Range of $f\left(x\right) = \left\{\frac{\pi }{4}, \frac{3\pi }{4}\right\}$