Question

If $f\left(x\right)={\sin }^{-1}x+{\sec }^{-1}x$ is defined, then which of the following value/values is/are in its range?

• A. $\frac{-\pi }{2}$
• B. $\frac{\pi }{2}$
• C. $\pi$
• D. $\frac{3\pi }{2}$

Answer 1

Answer: (B)

$\frac{\pi }{2}$

The domain for ${\sec }^{-1}\left(x\right)$ is $(-\infty ,-1]\cup [1,\infty )$

The domain for ${\sin }^{-1}\left(x\right)$ is $[-1,1]$

Hence the domain for

$f\left(x\right)={\sin }^{-1}\left(x\right)+{\sec }^{-1}\left(x\right)$ will be

$-1$ and $1$.

Therefore the range of values of $f\left(x\right)$ will be

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

$=\frac{\pi }{2}$

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

Hence the range of $f\left(x\right)$ includes a single element that is $\frac{\pi }{2}$