To solve this problem, we need to apply two transformations to the parent tangent function, [tex]\( \tan(x) \)[/tex]:
1. Horizontal Compression by a factor of [tex]\(\frac{1}{2}\)[/tex]:
- A horizontal compression by a factor of [tex]\(\frac{1}{2}\)[/tex] means that we replace [tex]\(x\)[/tex] with [tex]\(2x\)[/tex] in the function. So, the function [tex]\( \tan(x) \)[/tex] becomes [tex]\( \tan(2x) \)[/tex].
2. Reflection over the [tex]\(x\)[/tex]-axis:
- Reflecting a function over the [tex]\(x\)[/tex]-axis involves multiplying the function by [tex]\(-1\)[/tex]. Thus, [tex]\( \tan(2x) \)[/tex] becomes [tex]\( -\tan(2x) \)[/tex].
Combining these two transformations, we conclude that the new function [tex]\( g(x) \)[/tex] is given by:
[tex]\[ g(x) = -\tan(2x) \][/tex]
Since the transformations lead us to [tex]\( g(x) = -\tan(2x) \)[/tex], the correct equation representing the function [tex]\( g \)[/tex] is found in option A.
Therefore, the correct answer is:
[tex]\[ A. \, g(x) = -\tan(2x) \][/tex]
