Statement 1 : The domain of the function `f(x)=sqrt(x-[x])" is "R^(+)`
Statement 2 : The domain of the function `sqrt(f(x)) is {x : f (x) ge 0}`.

Options

A. Statement-1 is True, Statement-2 is True and Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To solve the given problem, we need to analyze both statements regarding the function \( f(x) = \sqrt{x - [x]} \).

### Step 1: Analyze Statement 1
**Statement 1:** The domain of the function \( f(x) = \sqrt{x - [x]} \) is \( \mathbb{R}^+ \).

- The greatest integer function \( [x] \) gives the largest integer less than or equal to \( x \). Therefore, \( x - [x] \) represents the fractional part of \( x \), denoted as \( \{x\} \).
- The fractional part \( \{x\} \) is defined as \( x - [x] \) and lies in the interval \( [0, 1) \).
- Thus, \( f(x) = \sqrt{\{x\}} \) is defined when \( \{x\} \geq 0 \), which is true for all \( x \) in \( \mathbb{R} \) (including negative values and zero).

**Conclusion for Statement 1:** Since \( f(x) \) is defined for all real numbers, Statement 1 is **false**.

### Step 2: Analyze Statement 2
**Statement 2:** The domain of the function \( \sqrt{f(x)} \) is \( \{x : f(x) \geq 0\} \).

- The function \( f(x) = \sqrt{x - [x]} \) is always non-negative because it is a square root function. Therefore, \( f(x) \geq 0 \) for all \( x \).
- This means that \( \sqrt{f(x)} \) is also defined for all \( x \) in \( \mathbb{R} \).

**Conclusion for Statement 2:** Since \( f(x) \) is non-negative for all \( x \), Statement 2 is **true**.

### Final Conclusion
- Statement 1 is **false**.
- Statement 2 is **true**.

### Answer Options
Given the analysis, the correct option is **Option 4**: Statement 1 is false and Statement 2 is true.