To determine for which values of [tex]\( x \)[/tex] the expression [tex]\(\frac{\sqrt{3 x^2}}{\sqrt{4 x}}\)[/tex] is defined, we need to consider the domain of the expression, paying close attention to the conditions imposed by the square roots and the division.
1. Square Root Condition:
- For [tex]\(\sqrt{3 x^2}\)[/tex] to be defined, the expression inside the square root must be non-negative.
[tex]\[
3 x^2 \geq 0
\][/tex]
Since [tex]\( x^2 \)[/tex] is always non-negative for all real [tex]\( x \)[/tex], [tex]\( 3 x^2 \)[/tex] is also non-negative for all real [tex]\( x \)[/tex]. Therefore, there are no additional restrictions from [tex]\(\sqrt{3 x^2}\)[/tex].
- For [tex]\(\sqrt{4 x}\)[/tex] to be defined, the expression inside the square root must be non-negative.
[tex]\[
4 x \geq 0 \implies x \geq 0
\][/tex]
2. Division by Zero Condition:
- We also need to ensure the denominator is not zero because division by zero is undefined.
[tex]\[
\sqrt{4 x} \neq 0
\][/tex]
This means:
[tex]\[
4 x \neq 0 \implies x \neq 0
\][/tex]
Combining the conditions from the square roots and the division:
- From the square root [tex]\(\sqrt{4 x}\)[/tex]: [tex]\( x \geq 0 \)[/tex]
- From the division: [tex]\( x \neq 0 \)[/tex]
The combined condition is:
[tex]\[
x > 0
\][/tex]
Therefore, the expression [tex]\(\frac{\sqrt{3 x^2}}{\sqrt{4 x}}\)[/tex] is defined for:
[tex]\[
\boxed{x > 0}
\][/tex]
So, the correct choice from the given options is:
C. [tex]\(x > 0\)[/tex]
