The domain of the function $f(x)=\sqrt{log_{10}(\frac{5x-x^{2}}{4})}$ is:

Question

Accepted Answer

To determine 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 defined. This means:

1. The argument of the logarithm, $\frac{5x - x^2}{4}$, must be positive because the logarithm of a non-positive number is undefined in the real number system.
2. The expression inside the square root, $\log_{10}\left(\frac{5x - x^2}{4}\right)$, must be non-negative because the square root of a negative number is not defined in the real number system.

Let's analyze these conditions:

### Step 1: Ensure $\frac{5x - x^2}{4} > 0$

This inequality simplifies to:

$$ 5x - x^2 > 0 $$

Factoring gives:

$$ x(5 - x) > 0 $$

This inequality holds when $ x $ is in the interval $ (0, 5) $.

### Step 2: Ensure $\log_{10}\left(\frac{5x - x^2}{4}\right) \geq 0$

This condition implies:

$$ \frac{5x - x^2}{4} \geq 1 $$

Multiplying both sides by 4 gives:

$$ 5x - x^2 \geq 4 $$

Rearranging terms, we get:

$$ x^2 - 5x + 4 \leq 0 $$

Factoring the quadratic expression:

$$ (x - 1)(x - 4) \leq 0 $$

This inequality holds for $ x $ in the interval $[1, 4]$.

### Conclusion

The domain of the function $ f(x) $ is the intersection of the intervals from Step 1 and Step 2. The intersection of $ (0, 5) $ and $[1, 4]$ is $[1, 4]$.

Therefore, the correct domain of the function is $[1, 4]$, which corresponds to Option 2.

### Explanation of Incorrect Options

- **Option 1: [0, 5]**: This option is incorrect because at $ x = 0 $, the expression $\frac{5x - x^2}{4}$ becomes 0, making the logarithm undefined. Additionally, the interval [0, 5] includes values outside the valid range determined by the inequality $(x - 1)(x - 4) \leq 0$.

- **Option 3: [1, 2]**: This option is incorrect because it does not cover the entire range where $(x - 1)(x - 4) \leq 0$ holds true. The valid domain extends up to 4, not just 2.

- **Option 4: [0, 1]**: This option is incorrect because it includes $ x = 0 $, where the expression inside the logarithm is 0, making the logarithm undefined. Additionally, it does not cover the full range of valid $ x $ values from 1 to 4.

Thus, Option 2 is the only correct choice for the domain of the function.