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The parent tangent function is horizontally compressed by a factor of [tex]\frac{1}{2}[/tex] and reflected over the [tex]x[/tex]-axis. Which equation could represent function [tex]g[/tex], the result of this transformation?

A. [tex]g(x)=-\tan(2x)[/tex]
B. [tex]g(x)=\tan(-2x)[/tex]
C. [tex]g(x)=-\tan\left(\frac{1}{2}x\right)[/tex]
D. [tex]g(x)=\tan\left(-\frac{1}{2}x\right)[/tex]

Sagot :

GinnyAnsweridn
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]
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