For x in mathbb{R},if f(x) = sqrt{log_{10}lef | vedclass

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Question

(B) For the function $f(x) = \sqrt{\log_{10}\left(\frac{3-x}{x}\right)}$ to be defined,the expression inside the square root must be non-negative and

Vedclass · Accepted Answer

$\left(0, \frac{3}{2}\right]$

Vedclass · Answer

(B) For the function $f(x) = \sqrt{\log_{10}\left(\frac{3-x}{x}\right)}$ to be defined,the expression inside the square root must be non-negative and the argument of the logarithm must be positive.$1$. Condition for the logarithm argument: $\frac{3-x}{x} > 0$.Solving this inequality,we get $x \in (0, 3)$.$2$. Condition for the square root: $\log_{10}\left(\frac{3-x}{x}\right) \geq 0$.This implies $\frac{3-x}{x} \geq 10^0$,so $\frac{3-x}{x} \geq 1$.$\frac{3-x}{x} - 1 \geq 0 \implies \frac{3-x-x}{x} \geq 0 \implies \frac{3-2x}{x} \geq 0$.Multiplying by $-1$ (and reversing the inequality),we get $\frac{2x-3}{x} \leq 0$.The critical points are $x = 0$ and $x = \frac{3}{2}$. Testing intervals,we find $x \in (0, \frac{3}{2}]$.Combining both conditions: $(0, 3) \cap (0, \frac{3}{2}] = (0, \frac{3}{2}]$.Thus,the domain of $f$ is $\left(0, \frac{3}{2}\right]$.

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

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