To determine which equation represents a valid transformation of the parent tangent function to obtain function [tex]\( m \)[/tex], we need to analyze each option. Let's go through them carefully.
A. [tex]\( g(x) = \tan(x) - \frac{\pi}{2} \)[/tex]
In this option, [tex]\(\frac{\pi}{2}\)[/tex] is subtracted from the entire tangent function. However, this does not represent a standard transformation like horizontal or vertical shifts or stretches/compressions. Instead, it results in a vertical shift of [tex]\(\frac{\pi}{2}\)[/tex] units, which is not generally associated with simple transformations of the tangent function.
B. [tex]\( g(x) = \tan(x + \pi) \)[/tex]
Here, [tex]\(\pi\)[/tex] is added inside the argument of the tangent function. This represents a horizontal shift but needs careful analysis:
[tex]\[
\tan(x + \pi) = \tan(x + \pi) = \tan(x)
\][/tex]
due to the period of the tangent function being [tex]\(\pi\)[/tex]. This is essentially the same function as [tex]\( \tan(x) \)[/tex].
C. [tex]\( g(x) = \tan(x - \pi) \)[/tex]
Similarly, this subtracts [tex]\(\pi\)[/tex] inside the argument of the tangent function. This also represents a horizontal shift:
[tex]\[
\tan(x - \pi) = \tan(x - \pi) = \tan(x)
\][/tex]
due to the periodicity of the tangent function being [tex]\(\pi\)[/tex]. This transformation is essentially the same as [tex]\( \tan(x) \)[/tex].
D. [tex]\( g(x) = \tan \left(x - \frac{\pi}{2}\right) \)[/tex]
This subtracts [tex]\(\frac{\pi}{2}\)[/tex] inside the argument of the tangent function, representing a horizontal shift of [tex]\(\frac{\pi}{2}\)[/tex] units to the right:
[tex]\[
\tan \left(x - \frac{\pi}{2}\right)
\][/tex]
This is a valid transformation of the tangent function, shifting the function to the right by [tex]\(\frac{\pi}{2}\)[/tex].
Based on the detailed analysis above, option D ([tex]\( g(x) = \tan \left(x - \frac{\pi}{2}\right) \)[/tex]) correctly represents function [tex]\( m \)[/tex] as a transformation of the parent tangent function.
