Let's break down each given option to determine which function has vertical asymptotes that correspond to the specified domain, specifically [tex]\( x \neq \frac{\pi}{4} + \frac{\pi}{2} n \)[/tex], where [tex]\( n \)[/tex] is an integer.
1. [tex]\( f(x) = \tan(2x - \pi) \)[/tex]
- The tangent function [tex]\( \tan(\theta) \)[/tex] has vertical asymptotes where [tex]\( \theta = \frac{(2k+1)\pi}{2} \)[/tex] for integers [tex]\( k \)[/tex].
- For [tex]\( f(x) \)[/tex]:
[tex]\[
2x - \pi = \frac{(2k+1) \pi}{2}
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
2x = \frac{(2k+1)\pi}{2} + \pi = \frac{(2k+1)\pi + 2\pi}{2} = \frac{(2k+3)\pi}{2}
\][/tex]
[tex]\[
x = \frac{(2k+3)\pi}{4}
\][/tex]
This does not match the given domain [tex]\( x \neq \frac{\pi}{4} + \frac{\pi}{2} n \)[/tex].
2. [tex]\( g(x) = \tan(x - \pi) \)[/tex]
- For [tex]\( g(x) \)[/tex]:
[tex]\[
x - \pi = \frac{(2k+1)\pi}{2}
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{(2k+1)\pi}{2} + \pi = \frac{(2k+1)\pi + 2\pi}{2} = \frac{(2k+3)\pi}{2}
\][/tex]
This does not match the given domain [tex]\( x \neq \frac{\pi}{4} + \frac{\pi}{2} n \)[/tex].
3. [tex]\( h(x) = \tan \left( x - \frac{\pi}{2} \right) \)[/tex]
- For [tex]\( h(x) \)[/tex]:
[tex]\[
x - \frac{\pi}{2} = \frac{(2k+1)\pi}{2}
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{(2k+1)\pi}{2} + \frac{\pi}{2} = \frac{(2k+2)\pi}{2} = (k+1) \pi
\][/tex]
Setting [tex]\( k = 2m \)[/tex] or [tex]\( k = 2m+1 \)[/tex], it becomes clear this matches [tex]\( x \neq \frac{\pi}{4} + \frac{\pi}{2} n \)[/tex], given the periodicity properties of tangent and matches the given domain restrictions.
4. [tex]\( j(x) = \tan \left( \frac{x}{2} - \pi \right) \)[/tex]
- For [tex]\( j(x) \)[/tex]:
[tex]\[
\frac{x}{2} - \pi = \frac{(2k+1)\pi}{2}
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
\frac{x}{2} = \frac{(2k+1)\pi}{2} + \pi = \frac{(2k+1)\pi + 2\pi}{2} = \frac{(2k+3)\pi}{2}
\][/tex]
[tex]\[
x = (2k+3)\pi
\][/tex]
This does not match the given domain [tex]\( x \neq \frac{\pi}{4} + \frac{\pi}{2} n \)[/tex].
Thus, the correct choice that represents a tangent function with a domain such that [tex]\( x \neq \frac{\pi}{4} + \frac{\pi}{2} n \)[/tex], where [tex]\( n \)[/tex] is an integer, is:
[tex]\[ h(x) = \tan \left( x - \frac{\pi}{2} \right) \][/tex]
So, the answer is [tex]\( \boxed{3} \)[/tex].
