The domain of definition of the function `f(x)=sqrt(log_(10) ((2-x)/(x))) ` is

Options

A. `(0, 1)`
B. `[0, 1 ]`
C. `(0, 1]`
D. `(0, 2 )`

Explanation

To find the domain of the function \( f(x) = \sqrt{\log_{10} \left( \frac{2-x}{x} \right)} \), we need to ensure that the expression inside the square root is non-negative, as the square root function is only defined for non-negative values.

### Step-by-Step Solution:

1. **Set Up the Inequality**:
We need to ensure that:
\[ \log_{10} \left( \frac{2-x}{x} \right) \geq 0 \]

2. **Convert the Logarithmic Inequality**:
The logarithm is non-negative when its argument is greater than or equal to 1:
\[ \frac{2-x}{x} \geq 1 \]

3. **Rearranging the Inequality**:
Multiply both sides by \( x \) (noting that \( x > 0 \) to avoid changing the direction of the inequality):
\[ 2 - x \geq x \]
This simplifies to:
\[ 2 \geq 2x \]
or
\[ x \leq 1 \]

4. **Consider the Denominator**:
Since \( x \) is in the denominator, we must also ensure that \( x \neq 0 \):
\[ x > 0 \]

5. **Combine the Conditions**:
We have two conditions:
- \( x \leq 1 \)
- \( x > 0 \)

Combining these gives:
\[ 0 < x \leq 1 \]

6. **Express the Domain**:
The domain of the function \( f(x) \) is:
\[ (0, 1] \]

### Final Answer:
The domain of the function \( f(x) = \sqrt{\log_{10} \left( \frac{2-x}{x} \right)} \) is \( (0, 1] \).