Question

For \( x \in \mathbb{R} \), if \( f(x) = \sqrt{\log_{10} \left( \frac{3-x}{x} \right)} \), then the domain of \( f \) is:

• A. \( \left[ 0, \frac{3}{2} \right] \)
• B. \( \left( 0, \frac{3}{2} \right] \)
• C. \( [0, 1] \)
• D. \( (0, 1) \)

Hint:

When working with logarithmic and square root functions, always ensure the argument is positive and the logarithm result is non-negative.

Answer 1

The Correct Option is B

For \( f(x) = \sqrt{\log_{10} \left( \frac{3-x}{x} \right)} \), the argument of the square root must be non-negative, and the logarithmic expression inside must be positive.
Step 1: The argument of the logarithm, \( \frac{3-x}{x} \), must be positive: \[ \frac{3-x}{x}>0 \] Solving this inequality, we find that \( x \in (0, 3) \).
Step 2: The value inside the logarithm must also satisfy the condition that the logarithm is non-negative: \[ \log_{10} \left( \frac{3-x}{x} \right) \geq 0 \] This implies: \[ \frac{3-x}{x} \geq 1 \] Solving this, we find \( x \in (0, \frac{3}{2}] \).
Thus, the domain of the function is \( (0, \frac{3}{2}] \).