The domain of the function defined by f(x) = sin(−1√(x−1)) is

• A: [1, 2]
• B: [−1, 1]
• C: [0, 1]
• D: None of these

To determine the domain of the function \( f(x) = \sin^{-1}(\sqrt{x-1}) \), we need to consider the conditions under which the expression is valid.

Understanding the Function

The function involves the inverse sine function, which has specific input restrictions. The argument of the inverse sine function must be between -1 and 1, inclusive.

Finding the Valid Range

First, we need to ensure that \( \sqrt{x-1} \) falls within this range:

• Since \( \sqrt{x-1} \) is always non-negative, we only need to consider the upper limit: \( \sqrt{x-1} \leq 1 \).
• Squaring both sides gives us \( x - 1 \leq 1 \), which simplifies to \( x \leq 2 \).

Considering the Square Root

Additionally, the expression under the square root must be non-negative:

• This means \( x - 1 \geq 0 \), or \( x \geq 1 \).

Combining the Conditions

From the two conditions, we find that:

• The lower limit is \( x \geq 1 \).
• The upper limit is \( x \leq 2 \).

Final Domain

Thus, the domain of the function is the interval \( [1, 2] \).

The correct answer is A: [1, 2].