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

Options

A. `x in(-oo, oo)`
B. `x in(0, oo)`
C. `x in(6, 16)`
D. `x in [0, oo)`

Answer

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

### Step 1: Analyze the Inner Function
The inner function is \( \sqrt{x} \). The square root function is defined for non-negative values, which means:
\[ x \geq 0 \]

### Step 2: Analyze the Absolute Value
Next, we consider the absolute value \( |\sqrt{x}| \). Since \( \sqrt{x} \) is non-negative for \( x \geq 0 \), we have:
\[ |\sqrt{x}| = \sqrt{x} \]
Thus, the expression simplifies to:
\[ f(x) = \sqrt{\sqrt{x}} \]

### Step 3: Analyze the Outer Function
Now, we need to ensure that the outer square root \( \sqrt{\sqrt{x}} \) is also defined. Since \( \sqrt{x} \) is non-negative for \( x \geq 0 \), the outer square root is also defined for:
\[ \sqrt{x} \geq 0 \]
This condition is satisfied for all \( x \geq 0 \).

### Conclusion
Therefore, the function \( f(x) = \sqrt{|\sqrt{x}|} \) is defined for all \( x \) such that:
\[ x \geq 0 \]
Thus, the domain of the function is:
\[ \text{Domain of } f(x) = [0, \infty) \]