The domain of the function `f(x)=sqrt(|x|)` is :

Options

A. `x in (-oo, oo)`
B. `x in (0, oo)`
C. `x in (-oo, 0)uu(0, oo)`
D. `x in [0, oo)`

Answer

To find the domain of the function \( f(x) = \sqrt{|x|} \), we need to determine the values of \( x \) for which the function returns real numbers.

### Step-by-Step Solution:

1. **Understanding the Function**: The function involves a square root, \( \sqrt{|x|} \). The square root function is defined for non-negative values. Therefore, we need to ensure that the expression inside the square root is non-negative.

2. **Analyzing the Absolute Value**: The absolute value \( |x| \) is always non-negative for any real number \( x \). This means that \( |x| \geq 0 \) for all \( x \in \mathbb{R} \).

3. **Setting Up the Inequality**: Since we need \( |x| \) to be greater than or equal to zero, we can write:
\[ |x| \geq 0 \]
This is true for all real numbers.

4. **Conclusion about the Domain**: Since \( |x| \) is non-negative for all real \( x \), the square root \( \sqrt{|x|} \) will also yield a real number for any real \( x \). Therefore, the domain of the function \( f(x) = \sqrt{|x|} \) is all real numbers.

5. **Final Result**: The domain of the function is:
\[ \text{Domain} = (-\infty, \infty) \]