To determine which \( x \)-value is in the domain of the function \( f(x) = 2 \cot(3x) + 4 \), we need to understand when the cotangent function is defined. The cotangent function, \(\cot(x)\), is undefined whenever \( x \) is an integer multiple of \(\pi\) because \(\cot(x) = \frac{\cos(x)}{\sin(x)}\) and sine equals zero at integer multiples of \(\pi\).
Thus, for the function \( \cot(3x) \) to be defined, \( 3x \) should not be an integer multiple of \(\pi\). Let’s examine each given \( x \) value:
1. \( x = \frac{\pi}{3} \)
[tex]\[
3x = 3 \left( \frac{\pi}{3} \right) = \pi
\][/tex]
Since \(\pi\) is an integer multiple of \(\pi\), \( \cot(3x) \) is undefined. So, \(\frac{\pi}{3}\) is not in the domain.
2. \( x = \frac{\pi}{4} \)
[tex]\[
3x = 3 \left( \frac{\pi}{4} \right) = \frac{3\pi}{4}
\][/tex]
Since \(\frac{3\pi}{4}\) is not an integer multiple of \(\pi\), \( \cot(3x) \) is defined. So, \(\frac{\pi}{4}\) is in the domain.
3. \( x = 2\pi \)
[tex]\[
3x = 3 (2\pi) = 6\pi
\][/tex]
Since \( 6\pi \) is an integer multiple of \(\pi\), \( \cot(3x) \) is undefined. So, \( 2\pi \) is not in the domain.
4. \( x = \pi \)
[tex]\[
3x = 3 \pi
\][/tex]
Since \( 3\pi \) is an integer multiple of \(\pi\), \( \cot(3x) \) is undefined. So, \( \pi \) is not in the domain.
From analyzing the given values, we find that the \( x \)-value in the domain of the function \( f(x) = 2 \cot(3x) + 4 \) is:
[tex]\[
\boxed{\frac{\pi}{4}}
\][/tex]
