The domain of the function defined by `f(x) = sin^(-1)sqrt(x-1)` is

Options

A. `[1,2]`
B. `[-1,1]`
C. `[0,1]`
D. None of these

Explanation

To find the domain of the function \( f(x) = \sin^{-1}(\sqrt{x-1}) \), we need to ensure that the expression inside the inverse sine function is valid. Here are the steps to determine the domain:

### Step 1: Identify the conditions for the square root
The expression \( \sqrt{x-1} \) is defined when the argument \( x-1 \) is non-negative. Therefore, we need:
\[ x - 1 \geq 0 \]
This simplifies to:
\[ x \geq 1 \]

### Step 2: Identify the conditions for the inverse sine function
The range of the sine function is between -1 and 1. Therefore, for \( \sin^{-1}(y) \) to be defined, \( y \) must satisfy:
\[ -1 \leq \sqrt{x-1} \leq 1 \]
Since \( \sqrt{x-1} \) is always non-negative, we only need to consider the upper bound:
\[ \sqrt{x-1} \leq 1 \]
Squaring both sides gives:
\[ x - 1 \leq 1 \]
This simplifies to:
\[ x \leq 2 \]

### Step 3: Combine the conditions
Now we combine the two conditions we found:
1. \( x \geq 1 \)
2. \( x \leq 2 \)

Thus, the domain of the function \( f(x) \) is:
\[ 1 \leq x \leq 2 \]
or in interval notation:
\[ [1, 2] \]

### Conclusion
The domain of the function \( f(x) = \sin^{-1}(\sqrt{x-1}) \) is \( [1, 2] \).

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