Question

Range of $f(x)=si{n}^{-1}x+ta{n}^{-1}x+se{c}^{-1}x$ 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)

$f(x)=si{n}^{-1}x+ta{n}^{-1}x+se{c}^{-1}x$
Domain of $si{n}^{-1}x=[-1,1]$
Domain of $ta{n}^{-1}x=(-\infty ,\infty )$
Domain of $se{c}^{-1}=(-\infty ,\infty )-(-1,1)$
Domain of $f(x)=[-1,1]\cap (-\infty ,\infty )\cap [(-\infty ,\infty )-(-1,1)]$
$=\{-1,1\}$
Now $f(-1)=si{n}^{-1}+ta{n}^{-1}(-1)+se{c}^{-1}(-1)$
$=-\frac{\pi }{2}-\frac{\pi }{4}+\pi =\frac{\pi }{4}$
and $f\left(1\right)=si{n}^{-1}\left(1\right)+ta{n}^{-1}\left(1\right)+se{c}^{-1}\left(1\right)$
$=\frac{\pi }{2}+\frac{\pi }{4}+0$
$=\frac{3\pi }{4}$
Range of $f\left(x\right)=\left\{\frac{\pi }{4},\frac{3\pi }{4}\right\}$