Find the domains and ranges of the functions.
`(1)/(|x|-x)`

Answer

To find the domain and range of the function \( f(x) = \frac{1}{|x| - x} \), we will analyze the function step by step.

### Step 1: Define the function
The function is given as:
\[ f(x) = \frac{1}{|x| - x} \]

### Step 2: Analyze the expression \(|x| - x\)
The expression \(|x| - x\) behaves differently for positive and negative values of \(x\).

- **Case 1: When \(x \geq 0\)**
Here, \(|x| = x\). Thus,
\[ |x| - x = x - x = 0 \]
This means that \(f(x)\) is undefined for all \(x \geq 0\) since we cannot divide by zero.

- **Case 2: When \(x < 0\)**
Here, \(|x| = -x\). Thus,
\[ |x| - x = -x - x = -2x \]
Therefore, for \(x < 0\),
\[ f(x) = \frac{1}{-2x} \]

### Step 3: Determine the domain of \(f(x)\)
From the analysis:
- The function is undefined for \(x \geq 0\).
- The function is defined for \(x < 0\).

Thus, the domain of \(f(x)\) is:
\[ \text{Domain} = (-\infty, 0) \]

### Step 4: Determine the range of \(f(x)\)
Now we need to analyze the function \(f(x) = \frac{1}{-2x}\) for \(x < 0\).

- As \(x\) approaches \(0\) from the left (i.e., \(x \to 0^{-}\)), \(f(x) = \frac{1}{-2x}\) approaches \(-\infty\).
- As \(x\) approaches \(-\infty\), \(f(x)\) approaches \(0\) from the negative side (i.e., \(f(x) \to 0^{-}\)).

Thus, the range of \(f(x)\) is:
\[ \text{Range} = (-\infty, 0) \]

### Final Answer
- **Domain**: \((- \infty, 0)\)
- **Range**: \((- \infty, 0)\)

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