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 defined as long as the denominator is not equal to zero. Therefore, we need to solve the inequality:

\[ x - |x| \neq 0 \]

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

The absolute value function \( |x| \) behaves differently depending on whether \( x \) is positive or negative. We will consider two cases based on the value of \( x \).

### Step 2: Case 1 - When \( x \geq 0 \)

If \( x \) is greater than or equal to zero, then \( |x| = x \). Thus, we can rewrite the expression:

\[ x - |x| = x - x = 0 \]

In this case, \( x - |x| = 0 \), which means the function is undefined for all \( x \geq 0 \).

### Step 3: Case 2 - When \( x < 0 \)

If \( x \) is less than zero, then \( |x| = -x \). Therefore, we can rewrite the expression as:

\[ x - |x| = x - (-x) = x + x = 2x \]

Now we need to check when this expression is not equal to zero:

\[ 2x \neq 0 \implies x \neq 0 \]

Since we are already considering the case where \( x < 0 \), this condition is satisfied for all \( x < 0 \).

### Step 4: Conclusion

From our analysis, we find that the function \( f(x) \) is undefined for \( x \geq 0 \) and is defined for all \( x < 0 \). Therefore, the domain of \( f(x) \) is:

\[ \text{Domain} = (-\infty, 0) \]