Find the domain of following functions:
`f(x)=1/(sqrt(abs(x)-x^(2)))`

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{|x| - x^2}} \), we need to ensure that the expression inside the square root is positive, as the square root must be defined and cannot be zero (since it is in the denominator).

### Step 1: Set up the inequality
We start with the condition that the expression under the square root must be greater than zero:
\[ |x| - x^2 > 0 \]

### Step 2: Analyze the expression
We can rewrite the inequality:
\[ |x| > x^2 \]
Now, we will consider two cases based on the definition of absolute value.

### Case 1: \( x \geq 0 \)
If \( x \geq 0 \), then \( |x| = x \). The inequality becomes:
\[ x > x^2 \]
Rearranging gives:
\[ x^2 - x < 0 \]
Factoring the left-hand side:
\[ x(x - 1) < 0 \]
This inequality holds true when \( x \) is between the roots \( 0 \) and \( 1 \):
\[ 0 < x < 1 \]

### Case 2: \( x < 0 \)
If \( x < 0 \), then \( |x| = -x \). The inequality becomes:
\[ -x > x^2 \]
Rearranging gives:
\[ x^2 + x < 0 \]
Factoring the left-hand side:
\[ x(x + 1) < 0 \]
This inequality holds true when \( x \) is between the roots \( -1 \) and \( 0 \):
\[ -1 < x < 0 \]

### Step 3: Combine the intervals
From both cases, we have:
1. From Case 1: \( 0 < x < 1 \)
2. From Case 2: \( -1 < x < 0 \)

Combining these intervals gives us:
\[ (-1, 0) \cup (0, 1) \]

### Step 4: Conclusion
The domain of the function \( f(x) = \frac{1}{\sqrt{|x| - x^2}} \) is:
\[ (-1, 0) \cup (0, 1) \]