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

Explanation

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, as well as ensuring that the logarithm is defined. Here’s a step-by-step solution:

### Step 1: Set up the inequality for the logarithm
The logarithm function \( \log_{1/4}(y) \) is defined for \( y > 0 \). Therefore, we need:
\[ \frac{5x - x^2}{4} > 0 \]
This simplifies to:
\[ 5x - x^2 > 0 \]

### Step 2: Factor the quadratic expression
Rearranging the inequality gives:
\[ -x^2 + 5x > 0 \]
Factoring out \(-1\):
\[ -x(x - 5) > 0 \]
This means:
\[ x(x - 5) < 0 \]

### Step 3: Determine the intervals
The critical points from the factored inequality are \( x = 0 \) and \( x = 5 \). We will test the intervals determined by these points: \( (-\infty, 0) \), \( (0, 5) \), and \( (5, \infty) \).

- For \( x < 0 \) (e.g., \( x = -1 \)): \( (-1)(-1 - 5) = (-1)(-6) = 6 > 0 \) (not in the solution set)
- For \( 0 < x < 5 \) (e.g., \( x = 1 \)): \( (1)(1 - 5) = (1)(-4) = -4 < 0 \) (in the solution set)
- For \( x > 5 \) (e.g., \( x = 6 \)): \( (6)(6 - 5) = (6)(1) = 6 > 0 \) (not in the solution set)

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

### Step 4: Set up the inequality for the logarithm to be non-negative
Next, we need to ensure that:
\[ \log_{1/4}\left(\frac{5x - x^2}{4}\right) \geq 0 \]
Since the base \( \frac{1}{4} < 1 \), the logarithm is non-negative when:
\[ \frac{5x - x^2}{4} \leq 1 \]
This simplifies to:
\[ 5x - x^2 \leq 4 \]
Rearranging gives:
\[ -x^2 + 5x - 4 \leq 0 \]
Factoring:
\[ -(x^2 - 5x + 4) \leq 0 \]
This means:
\[ x^2 - 5x + 4 \geq 0 \]

### Step 5: Factor the quadratic expression
Factoring the quadratic:
\[ (x - 1)(x - 4) \geq 0 \]

### Step 6: Determine the intervals
The critical points are \( x = 1 \) and \( x = 4 \). Testing intervals:
- For \( x < 1 \) (e.g., \( x = 0 \)): \( (0 - 1)(0 - 4) = ( -1)( -4) = 4 \geq 0 \) (in the solution set)
- For \( 1 < x < 4 \) (e.g., \( x = 2 \)): \( (2 - 1)(2 - 4) = (1)(-2) = -2 < 0 \) (not in the solution set)
- For \( x > 4 \) (e.g., \( x = 5 \)): \( (5 - 1)(5 - 4) = (4)(1) = 4 \geq 0 \) (in the solution set)

Thus, the solution to \( (x - 1)(x - 4) \geq 0 \) is:
\[ x \leq 1 \quad \text{or} \quad x \geq 4 \]

### Step 7: Find the intersection of the two conditions
We need to find the intersection of the two intervals:
1. From \( 0 < x < 5 \)
2. From \( x \leq 1 \) or \( x \geq 4 \)

The intersection is:
\[ (0, 1] \cup [4, 5) \]

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