Question

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

• A. $\{\frac{\pi }{4},\frac{3\pi }{5}\}$
• B. $\{\frac{\pi }{4},\frac{3\pi }{2}\}$
• C. $\{\frac{\pi }{4},\frac{3\pi }{4}\}$
• D. none of these

Answer 1

Answer: (C)

$\{\frac{\pi }{4},\frac{3\pi }{4}\}$

Domain of each inverse trigonometric term

$si{n}^{-1}x$ is $\left|x\right|\le 1$

$se{c}^{-1}x$ is $\left|x\right|\ge 1$

$ta{n}^{-1}x$ is $xϵR$

common domain is $\{-1,1\}$

$f\left(1\right)=\frac{\pi }{2}+\frac{\pi }{4}+0=\frac{3\pi }{4}$

$f(1-)=\frac{-\pi }{2}+\frac{-\pi }{4}+\pi =\frac{\pi }{4}$

so range of given function is $\{\frac{\pi }{4},\frac{3\pi }{4}\}$