To reflect the graph of [tex]\( f(x) = 3^x \)[/tex] over the [tex]\( x \)[/tex]-axis, we need to multiply the function by [tex]\(-1\)[/tex]. The steps involved are as follows:
1. Original Function: Start with the original function [tex]\( f(x) = 3^x \)[/tex].
2. Reflection Over the [tex]\( x \)[/tex]-Axis: When reflecting a function [tex]\( f(x) \)[/tex] over the [tex]\( x \)[/tex]-axis, the resulting function [tex]\( g(x) \)[/tex] is given by [tex]\( g(x) = -f(x) \)[/tex].
3. Substitute [tex]\( f(x) \)[/tex]: Substitute [tex]\( f(x) = 3^x \)[/tex] into the reflection equation:
[tex]\[
g(x) = -f(x) = -3^x
\][/tex]
Thus, the equation of the new graph after reflecting [tex]\( f(x) = 3^x \)[/tex] over the [tex]\( x \)[/tex]-axis is:
[tex]\[
g(x) = -3^x
\][/tex]
Now, let's identify the correct option from the given choices:
A. [tex]\( g(x) = -(3)^x \)[/tex]\
B. [tex]\( g(x) = 3^{-x} \)[/tex]\
C. [tex]\( g(x) = -\left(\frac{1}{3}\right)^x \)[/tex]\
D. [tex]\( g(x) = \left(\frac{1}{3}\right)^x \)[/tex]
Option A, [tex]\( g(x) = -(3)^x \)[/tex], correctly represents the reflection of [tex]\( 3^x \)[/tex] over the [tex]\( x \)[/tex]-axis. Therefore, the correct answer is:
A. [tex]\( g(x) = -(3)^x \)[/tex]
