To determine which equation could be used to represent function [tex]\( m \)[/tex], let's analyze the transformations presented in the given equations.
We begin with the parent tangent function, which is [tex]\( \tan(x) \)[/tex].
First, consider option A:
[tex]\[ g(x) = \tan (x - \pi) \][/tex]
This equation involves a horizontal shift to the right by [tex]\(\pi\)[/tex] units. The general form for a horizontal shift is [tex]\( \tan(x - c) \)[/tex], where [tex]\( c \)[/tex] is the shift amount.
Next, let's look at option B:
[tex]\[ g(x) = \tan(x) - \frac{\pi}{2} \][/tex]
This equation appears to involve a vertical shift. However, the tangent function has a period of [tex]\( \pi \)[/tex]. A vertical shift of [tex]\(-\frac{\pi}{2}\)[/tex] does not accurately represent a standard transformation on the tangent function in terms of typical behavior like periods, amplitude, or phase shifts. Therefore, this is not a common or recognizable transformation.
Option C states:
[tex]\[ g(x) = \tan(x + \pi) \][/tex]
This option represents a horizontal shift to the left by [tex]\(\pi\)[/tex] units. It's worth noting that [tex]\(\tan(x + \pi)\)[/tex] and [tex]\(\tan(x)\)[/tex] are equivalent due to the periodicity of the tangent function (since [tex]\( \tan(x + \pi) = \tan(x) \)[/tex]). Therefore, this does not fundamentally alter the function.
Finally, option D is:
[tex]\[ g(x) = \tan \left( x - \frac{\pi}{2} \right) \][/tex]
This equation describes a horizontal shift to the right by [tex]\(\frac{\pi}{2}\)[/tex] units. The tangent function exhibits vertical asymptotes and periodic behavior with respect to such shifts, so we expect the resulting function to have asymptotes adjusted accordingly to reflect this horizontal translation.
Analyzing each option , the most straightforward transformation matching how tangent function shifts should yield modified behavior. From here, options A and D stand out because of their clear horizontal shifts directly affecting the asymptote positions.
Therefore, the correct equation that could be used to represent function [tex]\( m \)[/tex] is:
D. [tex]\( g(x) = \tan \left(x - \frac{\pi}{2}\right) \)[/tex]
