Question

The domain of the function defined by f(x) = \(\cos^{-1}\sqrt{x-1}\) is

• A. [1, 2]
• B. [0, 2]
• C. [-1, 1]
• D. [0, 1]

Answer 1

The Correct Option is A

We are given the function \(f(x) = \cos^{-1}(\sqrt{x-1})\) and need to find its domain.

There are two conditions that must be satisfied for the function to be defined:

1. The expression inside the square root must be non-negative: \(x - 1 \ge 0 \implies x \ge 1\)
2. The argument of the inverse cosine function must be between -1 and 1 (inclusive): \(-1 \le \sqrt{x-1} \le 1\)

Since \(\sqrt{x-1}\) is always non-negative, we can rewrite the second condition as:

\(0 \le \sqrt{x-1} \le 1\)

Squaring all parts, we get:

\(0 \le x - 1 \le 1\)

Adding 1 to all parts, we get:

\(1 \le x \le 2\)

Combining both conditions \(x \ge 1\) and \(1 \le x \le 2\), we find that the domain is \(1 \le x \le 2\).

Thus, the correct option is (A) \([1, 2]\).