Question

Range of function \( \cos^{-1} x \) is:

• A. \( [0, \pi] - \{ \frac{\pi}{2} \} \)
• B. \( (0, \pi) \)
• C. \( (-\frac{\pi}{2}, \frac{\pi}{2}) - \{0\} \)
• D. \( [0, \pi] \)

Hint:

Remember the "CO" family: Range of \( \cos^{-1} x \), \( \cot^{-1} x \), and \( \sec^{-1} x \) are all related to the interval \( [0, \pi] \), while \( \sin^{-1} x \), \( \tan^{-1} x \), and \( \operatorname{cosec}^{-1} x \) are related to \( [-\pi/2, \pi/2] \).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The range of an inverse trigonometric function corresponds to its Principal Value Branch.
For a function to have an inverse, it must be bijective (one-to-one and onto).
Step 2: Detailed Explanation:
The cosine function \( f(x) = \cos x \) is not one-to-one over its entire domain \( \mathbb{R} \).
To define its inverse, we restrict the domain of \( \cos x \) to \( [0, \pi] \), where the function is strictly decreasing and covers all values from \( -1 \) to \( 1 \).
Therefore, the inverse function \( \cos^{-1} x \) has a domain of \( [-1, 1] \).
The set of values it produces (the range) is the interval \( [0, \pi] \).
Step 3: Final Answer:
The principal value branch or range of \( \cos^{-1} x \) is \( [0, \pi] \).