Question

Consider the following statements
Statement I ${\sec }^{-1}$ can be defined as a function, if the domain is $R$ and range is $[0,\pi ]-\left\{\frac{\pi }{2}\right\}$.
Statement II The principal value branch of the ${\sec }^{-1}$ is $[0,\pi ]-\left\{\frac{\pi }{2}\right\}$.
Choose the correct option.

• A. Only I is true
• B. Only II is true
• C. Both I and II are true
• D. Neither I nor II is true

Answer 1

Answer: (B)

${\sec }^{-1}$ can be defined as a function whose domain is $R-(-1,1)$ and range could be any of the intervals $[-\pi ,0]-\left\{\frac{-\pi }{2}\right\},[0,\pi ]-\left\{\frac{\pi }{2}\right\},[\pi ,2\pi ]-\left\{\frac{3\pi }{2}\right\}$, etc.
Corresponding to each of these intervals, we get different branches of the function ${\sec }^{-1}$.
The branch with range $[0,\pi ]-\left\{\frac{\pi }{2}\right\}$ is called the principal value branch of the function ${\sec }^{-1}$.