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 $\sec ^{-1}$.