To determine the equation for a cosecant function that has vertical asymptotes at \( x = \frac{\pi}{3} + \frac{\pi}{3} n \), where \( n \) is an integer, follow these steps:
1. Identify the general form of the vertical asymptotes for the cosecant function:
- The general form for the vertical asymptotes of the cosecant function \( \csc(kx) \) is given by \( x = \frac{n\pi}{k} \), where \( n \) is an integer.
2. Compare the given vertical asymptotes to the general form:
- For the given asymptotes, \( x = \frac{\pi}{3} + \frac{\pi}{3} n \).
- Rewrite this as \( x = \frac{\pi}{3}(1 + n) \). Here, \( 1 + n \) is also an integer, so this matches the form \( x = \frac{m\pi}{3} \) for integer \( m \).
3. Determine the value of \( k \) that fits the general form:
- Comparing \( x = \frac{m\pi}{3} \) with \( x = \frac{n\pi}{k} \), it is clear that \( k \) must equal 3 for the vertical asymptotes to match.
4. Match the value of \( k \) to one of the given function forms:
- Check the given options:
1. \( h(x) = 4 \csc x \): This would have vertical asymptotes at \( x = n\pi \), which does not match \( x = \frac{\pi}{3} + \frac{\pi}{3} n \).
2. \( g(x) = 3 \csc 2x \): This would have vertical asymptotes at \( x = \frac{n\pi}{2} \), which does not match.
3. \( f(x) = 3 \csc 4x \): This would have vertical asymptotes at \( x = \frac{n\pi}{4} \), which does not match.
4. \( j(x) = 4 \csc 3x \): This would have vertical asymptotes at \( x = \frac{n\pi}{3} \), which agrees with the given form.
Thus, the correct equation with vertical asymptotes at \( x = \frac{\pi}{3} + \frac{\pi}{3} n \) is:
[tex]\[ j(x) = 4 \csc 3x \][/tex]
