To find the domain of the function \( f(x) = \sqrt{\log_{10} \left( \frac{5x - x^2}{4} \right)} \), 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
The first condition we need to satisfy is:
\[
\frac{5x - x^2}{4} > 0
\]
This simplifies to:
\[
5x - x^2 > 0
\]
Multiplying both sides by 4 (which is positive and does not change the inequality):
\[
5x - x^2 > 0
\]
### Step 2: Factor the inequality
Rearranging gives:
\[
-x^2 + 5x > 0
\]
Factoring out \( -1 \):
\[
-x(x - 5) > 0
\]
Multiplying through by -1 (which reverses the inequality):
\[
x(x - 5) < 0
\]
### Step 3: Find critical points
The critical points from the factors are:
\[
x = 0 \quad \text{and} \quad x = 5
\]
We will test the intervals determined by these points: \( (-\infty, 0) \), \( (0, 5) \), and \( (5, \infty) \).
### Step 4: Test intervals
1. **Interval \( (-\infty, 0) \)**: Choose \( x = -1 \):
\[
(-1)(-1 - 5) = (-1)(-6) = 6 > 0 \quad \text{(not in the domain)}
\]
2. **Interval \( (0, 5) \)**: Choose \( x = 1 \):
\[
(1)(1 - 5) = (1)(-4) = -4 < 0 \quad \text{(in the domain)}
\]
3. **Interval \( (5, \infty) \)**: Choose \( x = 6 \):
\[
(6)(6 - 5) = (6)(1) = 6 > 0 \quad \text{(not in the domain)}
\]
Thus, the solution to \( x(x - 5) < 0 \) is:
\[
0 < x < 5
\]
### Step 5: Ensure the logarithm is defined
Next, we need to ensure that:
\[
\log_{10} \left( \frac{5x - x^2}{4} \right) \geq 0
\]
This means:
\[
\frac{5x - x^2}{4} \geq 1
\]
Multiplying both sides by 4:
\[
5x - x^2 \geq 4
\]
Rearranging gives:
\[
-x^2 + 5x - 4 \geq 0
\]
Factoring:
\[
-(x^2 - 5x + 4) \geq 0
\]
Factoring the quadratic:
\[
-(x - 1)(x - 4) \geq 0
\]
### Step 6: Find critical points for the logarithm
The critical points are:
\[
x = 1 \quad \text{and} \quad x = 4
\]
We will test the intervals \( (-\infty, 1) \), \( (1, 4) \), and \( (4, \infty) \).
### Step 7: Test intervals for logarithm
1. **Interval \( (-\infty, 1) \)**: Choose \( x = 0 \):
\[
-(0 - 1)(0 - 4) = -(-1)(-4) = -4 < 0 \quad \text{(not in the domain)}
\]
2. **Interval \( (1, 4) \)**: Choose \( x = 2 \):
\[
-(2 - 1)(2 - 4) = -1(-2) = 2 > 0 \quad \text{(in the domain)}
\]
3. **Interval \( (4, \infty) \)**: Choose \( x = 5 \):
\[
-(5 - 1)(5 - 4) = -4(1) = -4 < 0 \quad \text{(not in the domain)}
\]
Thus, the solution to \( -(x - 1)(x - 4) \geq 0 \) is:
\[
1 \leq x \leq 4
\]
### Step 8: Find the intersection of the two conditions
The first condition gives \( 0 < x < 5 \) and the second condition gives \( 1 \leq x \leq 4 \). The intersection of these two intervals is:
\[
1 \leq x < 4
\]
### Final Domain
Thus, the domain of the function \( f(x) \) is:
\[
\boxed{[1, 4)}
\]
