Domain of the function, `f(x)=[log_10 ((5x-x^2)/4)]^(1/2)` is

Options

A. `-oo lt x lt oo`
B. `1 le x le 4 `
C. `4 le x le 16`
D. ` -1 le x le 1`

Explanation

To find the domain of the function \( f(x) = \left[ \log_{10} \left( \frac{5x - x^2}{4} \right) \right]^{1/2} \), we need to ensure that the expression inside the logarithm is positive and that the logarithm itself is non-negative (since we are taking the square root).

### Step-by-Step Solution:

1. **Identify the condition for the logarithm:**
The logarithm \( \log_{10} \left( \frac{5x - x^2}{4} \right) \) is defined when its argument is positive:
\[ \frac{5x - x^2}{4} > 0 \]
This simplifies to:
\[ 5x - x^2 > 0 \]
or equivalently:
\[ x(5 - x) > 0 \]

2. **Solve the inequality \( x(5 - x) > 0 \):**
The expression \( x(5 - x) \) is a quadratic function that opens downwards. To find the intervals where it is positive, we first find the roots:
\[ x = 0 \quad \text{and} \quad x = 5 \]
Now we test the intervals determined by these roots: \( (-\infty, 0) \), \( (0, 5) \), and \( (5, \infty) \).

- For \( x < 0 \): Choose \( x = -1 \):
\[ -1(5 - (-1)) = -1(6) < 0 \quad \text{(not in the domain)} \]
- For \( 0 < x < 5 \): Choose \( x = 1 \):
\[ 1(5 - 1) = 1(4) > 0 \quad \text{(in the domain)} \]
- For \( x > 5 \): Choose \( x = 6 \):
\[ 6(5 - 6) = 6(-1) < 0 \quad \text{(not in the domain)} \]

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

3. **Consider the logarithm being non-negative:**
We also need:
\[ \log_{10} \left( \frac{5x - x^2}{4} \right) \geq 0 \]
This means:
\[ \frac{5x - x^2}{4} \geq 1 \]
Simplifying this gives:
\[ 5x - x^2 \geq 4 \]
or:
\[ x^2 - 5x + 4 \leq 0 \]

4. **Solve the quadratic inequality \( x^2 - 5x + 4 \leq 0 \):**
The roots of the quadratic equation \( x^2 - 5x + 4 = 0 \) are:
\[ x = 1 \quad \text{and} \quad x = 4 \]
Now we test the intervals \( (-\infty, 1) \), \( (1, 4) \), and \( (4, \infty) \).

- For \( x < 1 \): Choose \( x = 0 \):
\[ 0^2 - 5(0) + 4 = 4 > 0 \quad \text{(not in the domain)} \]
- For \( 1 < x < 4 \): Choose \( x = 2 \):
\[ 2^2 - 5(2) + 4 = 4 - 10 + 4 = -2 \leq 0 \quad \text{(in the domain)} \]
- For \( x > 4 \): Choose \( x = 5 \):
\[ 5^2 - 5(5) + 4 = 0 \quad \text{(boundary, included)} \]

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

5. **Combine the results:**
The domain of \( f(x) \) must satisfy both conditions:
- From \( x(5 - x) > 0 \): \( 0 < x < 5 \)
- From \( x^2 - 5x + 4 \leq 0 \): \( 1 \leq x \leq 4 \)

The intersection of these intervals is:
\[ 1 \leq x \leq 4 \]

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