The domain of the function `f (x)= sqrt(log_( 0.4) (x-x^2))` is

Options

A. `(1,2)`
B. `(0,1)`
C. `(0,1/2)`
D. None of these

Answer

To find the domain of the function \( f(x) = \sqrt{\log_{0.4}(x - x^2)} \), we need to ensure that the expression inside the square root is non-negative and that the logarithm is defined. Here are the steps to solve the problem:

### Step 1: Set the condition for the logarithm
The logarithm \( \log_{0.4}(x - x^2) \) is defined only when \( x - x^2 > 0 \). Therefore, we need to solve the inequality:
\[ x - x^2 > 0 \]

### Step 2: Factor the inequality
We can factor the expression:
\[ x(1 - x) > 0 \]
This means we need to find the values of \( x \) for which the product is positive.

### Step 3: Find the critical points
The critical points occur when \( x = 0 \) or \( x = 1 \). We will use these points to test the intervals:
- Interval 1: \( (-\infty, 0) \)
- Interval 2: \( (0, 1) \)
- Interval 3: \( (1, \infty) \)

### Step 4: Test the intervals
1. **For \( x < 0 \)** (e.g., \( x = -1 \)):
\[ -1(1 - (-1)) = -1(2) = -2 < 0 \quad \text{(not valid)} \]
2. **For \( 0 < x < 1 \)** (e.g., \( x = 0.5 \)):
\[ 0.5(1 - 0.5) = 0.5(0.5) = 0.25 > 0 \quad \text{(valid)} \]
3. **For \( x > 1 \)** (e.g., \( x = 2 \)):
\[ 2(1 - 2) = 2(-1) = -2 < 0 \quad \text{(not valid)} \]

### Step 5: Determine the valid interval
From the tests, we find that \( x(1 - x) > 0 \) is satisfied for:
\[ x \in (0, 1) \]

### Step 6: Set the condition for the logarithm to be non-negative
Next, we need \( \log_{0.4}(x - x^2) \geq 0 \). Since the base \( 0.4 < 1 \), the logarithm is non-negative when:
\[ x - x^2 \leq 1 \]

### Step 7: Solve the inequality
We can rearrange this as:
\[ x - x^2 - 1 \leq 0 \]
This can be rewritten as:
\[ -x^2 + x - 1 \leq 0 \]
or
\[ x^2 - x + 1 \geq 0 \]

### Step 8: Find the roots of the quadratic
The roots of the quadratic \( x^2 - x + 1 = 0 \) can be found using the discriminant:
\[ D = b^2 - 4ac = (-1)^2 - 4(1)(1) = 1 - 4 = -3 \]
Since the discriminant is negative, the quadratic \( x^2 - x + 1 \) is always positive. Therefore, this condition does not restrict \( x \).

### Step 9: Final domain
The only restriction comes from the first inequality, which gives us:
\[ x \in (0, 1) \]

Thus, the domain of the function \( f(x) = \sqrt{\log_{0.4}(x - x^2)} \) is:
\[ \boxed{(0, 1)} \]