The range of `f(x)=sqrt(|x|-x)` is:

Options

A. `(0,oo)`
B. `[0,oo)`
C. `(-oo,0)`
D. `(-oo,0]`

Answer

To find the range of the function \( f(x) = \sqrt{|x| - x} \), we will follow these steps:

### Step 1: Analyze the expression inside the square root
The expression inside the square root is \( |x| - x \). We need to determine how this expression behaves for different values of \( x \).

### Step 2: Break down the absolute value
The absolute value function \( |x| \) can be defined piecewise:
- If \( x \geq 0 \), then \( |x| = x \).
- If \( x < 0 \), then \( |x| = -x \).

### Step 3: Evaluate \( |x| - x \) for different cases
1. **Case 1: \( x \geq 0 \)**
\[ |x| - x = x - x = 0 \]
2. **Case 2: \( x < 0 \)**
\[ |x| - x = -x - x = -2x \]

### Step 4: Determine the values of \( f(x) \)
- For \( x \geq 0 \):
\[ f(x) = \sqrt{0} = 0 \]
- For \( x < 0 \):
\[ f(x) = \sqrt{-2x} \]
Since \( x \) is negative, \( -2x \) is positive, and as \( x \) approaches \( 0 \) from the left, \( -2x \) approaches \( 0 \). As \( x \) decreases (becomes more negative), \( -2x \) increases without bound.

### Step 5: Find the minimum and maximum values of \( f(x) \)
- The minimum value of \( f(x) \) occurs at \( x = 0 \), where \( f(0) = 0 \).
- The maximum value of \( f(x) \) occurs as \( x \) approaches \( -\infty \), where \( f(x) \) approaches \( \infty \).

### Step 6: Conclude the range of \( f(x) \)
Thus, the range of \( f(x) \) is:
\[ [0, \infty) \]

### Final Answer
The range of \( f(x) = \sqrt{|x| - x} \) is \( [0, \infty) \).

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