To solve this problem, we need to understand the effect of reflecting a function over the [tex]\( x \)[/tex]-axis. Specifically, we start with the function [tex]\( f(x) = 3^x \)[/tex] and determine its new form after reflection.
### Step-by-Step Solution:
1. Original Function: The original function given is [tex]\( f(x) = 3^x \)[/tex].
2. Reflection Over the [tex]\( x \)[/tex]-Axis:
- Reflecting a function [tex]\( f(x) \)[/tex] over the [tex]\( x \)[/tex]-axis involves multiplying the function by [tex]\(-1\)[/tex].
- Mathematically, if [tex]\( f(x) \)[/tex] is the original function, the reflected function [tex]\( g(x) \)[/tex] is given by:
[tex]\[
g(x) = -f(x)
\][/tex]
3. Apply Reflection to the Given Function:
- Since [tex]\( f(x) = 3^x \)[/tex], multiply it by [tex]\(-1\)[/tex] to get the new function:
[tex]\[
g(x) = -3^x
\][/tex]
4. Match with Given Options:
- We now compare the derived equation [tex]\( g(x) = -3^x \)[/tex] with the provided options:
[tex]\[
\begin{aligned}
\text{A. } & g(x) = -\left(\frac{1}{3}\right)^x \\
\text{B. } & g(x) = -(3)^x \\
\text{C. } & g(x) = \left(\frac{1}{3}\right)^x \\
\text{D. } & g(x) = 3^{-x}
\end{aligned}
\][/tex]
- Clearly, option B, [tex]\( g(x) = -(3)^x \)[/tex], matches our derived equation [tex]\( g(x) = -3^x \)[/tex].
5. Conclusion:
- The correct answer is B.
Therefore, if the graph of [tex]\( f(x)=3^x \)[/tex] is reflected over the [tex]\( x \)[/tex]-axis, the equation of the new graph is:
[tex]\[
\boxed{-(3)^x}
\][/tex]
