The domain of the function `sqrt(x^2 - [x]^2)`, where [x] has the usual meaning, is

Options

A. any +ive number
B. any-ive number
C. any +ive number or -ive integer
D. none

Answer

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

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

### Step 1: Understanding the Greatest Integer Function
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). For any \(x\), we can express it 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: Expressing \(x^2\) and \([x]^2\)
Substituting \(x\) into the inequality, we have:

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

And since \([x] = n\):

\[ [x]^2 = n^2 \]

### Step 3: Setting Up the Inequality
Now we can substitute these into the inequality:

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

This simplifies to:

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

### Step 4: Analyzing the Inequality
The expression \(2nf + f^2\) can be factored as:

\[ f(2n + f) \geq 0 \]

### Step 5: Finding Conditions for \(f(2n + f) \geq 0\)
1. **Case 1: \(f = 0\)**
If \(f = 0\), then \(x\) is an integer, and the inequality holds since \(0 \geq 0\).

2. **Case 2: \(f > 0\)**
Here, \(0 < f < 1\). For the product \(f(2n + f)\) to be non-negative:
- If \(n \geq 0\), \(2n + f > 0\) (always true).
- If \(n < 0\), \(2n + f\) can be negative. We need \(2n + f \geq 0\), which gives \(f \geq -2n\). Since \(0 < f < 1\), we need:

\[ -2n < 1 \implies n > -\frac{1}{2} \]

Since \(n\) is an integer, this means \(n \geq 0\).

### Step 6: Conclusion on the Domain
From the analysis:
- **Positive integers** and **zero** satisfy the condition.
- **Negative integers** do not satisfy the condition since \(f\) cannot be positive if \(n < 0\).

Thus, the domain of the function \(f(x) = \sqrt{x^2 - [x]^2}\) is:

\[ \text{Domain} = \{ x \in \mathbb{R} : x \text{ is a non-negative integer} \} \]