Question

The range of the function, $f( x )=\cot ^{-1} x +\sec ^{-1} x +\operatorname{cosec}^{-1} x$, is

• A. $\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$
• B. $\left(\frac{\pi}{2}, \frac{3 \pi}{4}\right] \cup\left[\frac{5 \pi}{4}, \frac{3 \pi}{2}\right)$
• C. $\left[\frac{\pi}{2}, \pi\right) \cup\left(\pi, \frac{3 \pi}{2}\right]$
• D. $\left(\frac{\pi}{2}, \pi\right) \cup\left(\pi, \frac{3 \pi}{2}\right)$

Answer 1

Answer: (B)

Domain of $f( x )$ is $| x | \geq 1$
From the graph for $x \geq 1 f ( x )$ can attain values from $\left(\frac{\pi}{2}, \frac{3 \pi}{4}\right]$
also for $x \leq-1, f(x)$ can attain values from $\left[\frac{5 \pi}{4}, \frac{3 \pi}{2}\right)$