Find the domain of `f(x)=(1)/(x-|x|)`, when `x in R`

Answer

To find the domain of the function \( f(x) = \frac{1}{x - |x|} \), we need to determine the values of \( x \) for which the function is defined. The function is undefined when the denominator equals zero, i.e., when \( x - |x| = 0 \).

### Step 1: Analyze the expression \( x - |x| \)

The absolute value function \( |x| \) behaves differently based on whether \( x \) is positive or negative. We will consider both cases:

1. **Case 1**: \( x \geq 0 \)
- Here, \( |x| = x \).
- Therefore, \( x - |x| = x - x = 0 \).

2. **Case 2**: \( x < 0 \)
- Here, \( |x| = -x \).
- Therefore, \( x - |x| = x - (-x) = x + x = 2x \).

### Step 2: Set the denominator to zero

Now, we need to find when \( x - |x| = 0 \):

- From **Case 1**: \( x - |x| = 0 \) when \( x \geq 0 \).
- From **Case 2**: \( 2x = 0 \) when \( x < 0 \) gives \( x = 0 \).

### Step 3: Determine the values of \( x \) that make the function undefined

The function \( f(x) \) is undefined when:
- \( x - |x| = 0 \) which occurs at \( x = 0 \).

### Step 4: Define the domain

Since the function is undefined at \( x = 0 \), the domain of \( f(x) \) is all real numbers except \( 0 \).

Thus, the domain can be expressed in interval notation as:
\[ \text{Domain of } f(x) = (-\infty, 0) \cup (0, \infty) \]

### Final Answer:
The domain of \( f(x) = \frac{1}{x - |x|} \) is \( (-\infty, 0) \cup (0, \infty) \).

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