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

Answer: (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 $f\left(x\right) = \left\{\frac{\pi }{4}, \frac{3\pi }{4}\right\}$