The domain of the function `f(x)=sqrt(x^2-[x]^2)` , where `[x]` is the greatest integer less than or equal to `x ,` is `R` (b) `[0,+oo]` `(-oo,0)` (d) none of these

Options

A. `R`
B. `[0, oo)`
C. `(-oo, 0]`
D. none of these

Explanation

To find the domain of the function \( f(x) = \sqrt{x^2 - [x]^2} \), where \([x]\) is the greatest integer less than or equal to \(x\), we need to ensure that the expression inside the square root is non-negative. This means we need to solve the inequality:

\[ x^2 - [x]^2 \geq 0 \]

### Step 1: Analyze the expression

The term \([x]\) represents the greatest integer less than or equal to \(x\). For any real number \(x\), we can express \(x\) as:

\[ x = n + f \]

where \(n = [x]\) (an integer) and \(f\) is the fractional part of \(x\) such that \(0 \leq f < 1\).

### Step 2: Substitute and simplify

Substituting \(x\) into the expression gives:

\[ x^2 = (n + f)^2 = n^2 + 2nf + f^2 \]
\[ [x]^2 = n^2 \]

Thus, we have:

\[ x^2 - [x]^2 = (n^2 + 2nf + f^2) - n^2 = 2nf + f^2 \]

### Step 3: Set up the inequality

We need to solve the inequality:

\[ 2nf + f^2 \geq 0 \]

### Step 4: Analyze cases

1. **Case 1: \(x\) is an integer**
If \(x = n\) (an integer), then \(f = 0\). Thus:

\[ 2n(0) + 0^2 = 0 \geq 0 \]

This is true for all integers \(n\).

2. **Case 2: \(x\) is not an integer**
Here, \(0 < f < 1\). Since \(n\) is the greatest integer less than \(x\), \(n\) is also less than \(x\), which means \(n\) can be positive or negative.

- If \(n \geq 0\) (non-negative integers), then \(2nf + f^2 \geq 0\) is always true since both terms are non-negative.
- If \(n < 0\) (negative integers), then \(2nf\) is negative and \(f^2\) is positive. The term \(2nf + f^2\) will be non-negative if \(f\) is sufficiently small. However, as \(f\) approaches 1, \(2nf\) will dominate and make the expression negative.

### Step 5: Conclusion on the domain

From the analysis:
- All integers \(n\) are included in the domain.
- For \(x > 0\), the function is valid for all positive real numbers.
- For \(x < 0\), the function is valid only for negative integers.

Thus, the domain of \(f(x)\) is:

\[ \text{Domain} = \{ x \in \mathbb{R} : x \text{ is an integer} \} \cup (0, +\infty) \]

### Final Answer

The correct option is: **(b) [0, +∞)**.

---