To find the domain and range of the function \( f(x) = \frac{1}{|x| - x} \), we will follow these steps:
### Step 1: Identify when the function is defined
The function is defined when the denominator is not equal to zero. Therefore, we need to solve the equation:
\[
|x| - x = 0
\]
### Step 2: Solve the equation
This simplifies to:
\[
|x| = x
\]
The absolute value \( |x| \) equals \( x \) when \( x \) is non-negative (i.e., \( x \geq 0 \)).
### Step 3: Consider the cases for \( x \)
1. If \( x \geq 0 \):
- Here, \( |x| = x \), so the equation holds true.
2. If \( x < 0 \):
- Here, \( |x| = -x \), so \( -x = x \) leads to \( 0 = 2x \), which gives \( x = 0 \). However, since \( x < 0 \) is assumed, this case does not contribute any valid solutions.
### Step 4: Determine the domain
From the analysis, we find that the function is undefined when \( x \geq 0 \) because the denominator becomes zero. Thus, the function is defined for:
\[
x < 0
\]
The domain of the function is:
\[
(-\infty, 0)
\]
### Step 5: Find the range of the function
To find the range, we express the function for \( x < 0 \):
\[
f(x) = \frac{1}{|x| - x} = \frac{1}{-x - x} = \frac{1}{-2x}
\]
Since \( x < 0 \), \( -2x > 0 \). Therefore, \( f(x) \) is positive.
### Step 6: Analyze the behavior of \( f(x) \)
As \( x \) approaches \( 0 \) from the left (i.e., \( x \to 0^- \)), \( -2x \to 0^+ \), which means \( f(x) \to +\infty \).
As \( x \) approaches \( -\infty \), \( -2x \to +\infty \), which means \( f(x) \to 0^+ \).
### Step 7: Conclude the range
Thus, the range of \( f(x) \) is:
\[
(0, +\infty)
\]
### Final Answer
- **Domain**: \( (-\infty, 0) \)
- **Range**: \( (0, +\infty) \)
