Find the domain of the function
`f(x) = sqrt(log_(1//4) ((5x - x^(2))/(4)))`

Answer

To find the domain of the function \( f(x) = \sqrt{\log_{1/4} \left(\frac{5x - x^2}{4}\right)} \), we need to ensure that the expression inside the square root is non-negative and that the argument of the logarithm is positive.

### Step 1: Ensure the argument of the logarithm is positive
We start with the condition for the logarithm:
\[ \frac{5x - x^2}{4} > 0 \]
This simplifies to:
\[ 5x - x^2 > 0 \]
Factoring gives:
\[ -x^2 + 5x > 0 \quad \text{or} \quad x(5 - x) > 0 \]
This inequality is satisfied when \( x \) is between the roots of the equation \( x(5 - x) = 0 \), which are \( x = 0 \) and \( x = 5 \).

### Step 2: Determine the intervals
To find the intervals where \( x(5 - x) > 0 \), we can test the intervals:
- For \( x < 0 \): \( x(5 - x) < 0 \)
- For \( 0 < x < 5 \): \( x(5 - x) > 0 \)
- For \( x > 5 \): \( x(5 - x) < 0 \)

Thus, the solution to \( x(5 - x) > 0 \) is:
\[ 0 < x < 5 \]

### Step 3: Ensure the logarithm is non-negative
Next, we need to ensure that the logarithm itself is non-negative:
\[ \log_{1/4} \left(\frac{5x - x^2}{4}\right) \geq 0 \]
Since the base \( \frac{1}{4} < 1 \), we can rewrite this condition as:
\[ \frac{5x - x^2}{4} \leq 1 \]
Multiplying through by 4 gives:
\[ 5x - x^2 \leq 4 \]
Rearranging leads to:
\[ -x^2 + 5x - 4 \leq 0 \]
Factoring gives:
\[ -(x^2 - 5x + 4) \leq 0 \quad \text{or} \quad x^2 - 5x + 4 \geq 0 \]
Finding the roots of \( x^2 - 5x + 4 = 0 \) using the quadratic formula:
\[ x = \frac{5 \pm \sqrt{(5)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = \frac{5 \pm \sqrt{9}}{2} = \frac{5 \pm 3}{2} \]
This gives us:
\[ x = 4 \quad \text{and} \quad x = 1 \]

### Step 4: Determine the intervals for the quadratic inequality
To find where \( x^2 - 5x + 4 \geq 0 \), we test the intervals:
- For \( x < 1 \): \( x^2 - 5x + 4 > 0 \)
- For \( 1 < x < 4 \): \( x^2 - 5x + 4 < 0 \)
- For \( x > 4 \): \( x^2 - 5x + 4 > 0 \)

Thus, the solution to \( x^2 - 5x + 4 \geq 0 \) is:
\[ (-\infty, 1] \cup [4, \infty) \]

### Step 5: Find the intersection of the intervals
Now we find the intersection of the two conditions:
1. From \( 0 < x < 5 \)
2. From \( (-\infty, 1] \cup [4, \infty) \)

The intersection gives:
- From \( 0 < x < 1 \)
- From \( 4 < x < 5 \)

Thus, the domain of the function is:
\[ (0, 1) \cup (4, 5) \]

### Final Answer
The domain of the function \( f(x) \) is:
\[ \boxed{(0, 1) \cup (4, 5)} \]