The domain of definition of the function
`f(x)=(1)/(sqrt(|x|+x))` is

Options

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

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{|x| + x}} \), we need to analyze the expression inside the square root and ensure that it is defined and positive, as the square root function is only defined for non-negative values, and the denominator cannot be zero.

### Step-by-Step Solution:

1. **Understand the Absolute Value**:
The expression \( |x| + x \) behaves differently based on whether \( x \) is positive or negative. We will consider two cases: when \( x \geq 0 \) and when \( x < 0 \).

2. **Case 1: \( x \geq 0 \)**:
- When \( x \) is non-negative, \( |x| = x \).
- Therefore, \( |x| + x = x + x = 2x \).
- The function becomes:
\[ f(x) = \frac{1}{\sqrt{2x}} \]
- For \( f(x) \) to be defined, \( 2x \) must be greater than 0:
\[ 2x > 0 \implies x > 0 \]
- Thus, for this case, \( x \) can take values in the interval \( (0, \infty) \).

3. **Case 2: \( x < 0 \)**:
- When \( x \) is negative, \( |x| = -x \).
- Therefore, \( |x| + x = -x + x = 0 \).
- The function becomes:
\[ f(x) = \frac{1}{\sqrt{0}} = \frac{1}{0} \]
- Since division by zero is undefined, \( f(x) \) is not defined for \( x < 0 \).

4. **Combine Results**:
- From Case 1, we found that \( f(x) \) is defined for \( x > 0 \).
- From Case 2, we found that \( f(x) \) is not defined for \( x < 0 \) or \( x = 0 \).

5. **Conclusion**:
- The domain of the function \( f(x) \) is \( (0, \infty) \).

### Final Answer:
The domain of the function \( f(x) = \frac{1}{\sqrt{|x| + x}} \) is \( (0, \infty) \).

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