To determine the correct equation for $g(x)$ in terms of $f(x)$, we need to analyze the given transformations from the graph of $y = f(x)$ to the graph of $y = g(x)$.

1. Vertical Stretch/Compression and Reflection:

   • The graph of $y = g(x)$ appears to be a reflection of $y = f(x)$ over the x-axis.
   • The graph also appears to be vertically compressed by a factor of $\frac{1}{2}$.

2. Horizontal Stretch/Compression and Reflection:

   • The graph of $y = g(x)$ is also horizontally stretched by a factor of 2 compared to $y = f(x)$.
   • The horizontal stretching means that every x-value in $g(x)$ corresponds to half the x-value in $f(x)$.

Considering these transformations, we need to find the transformation that combines a horizontal stretch by a factor of 2, a reflection over the x-axis, and a vertical compression by a factor of $\frac{1}{2}$.

Given these transformations, the correct equation is:

$g(x) = f\left(-\frac{1}{2}x\right)$

Therefore, the correct answer is A. $g(x) = f\left( -\frac{1}{2} x \right)$.

Would you like further details on any part of the explanation? Here are some related questions you might consider:

1. How does horizontal stretching/compression affect the graph of a function?
2. How do reflections over the x-axis and y-axis transform a function?
3. What are the general forms of transformations for functions?
4. How can you determine transformations from a graph visually?
5. Can you show a step-by-step process for verifying transformations algebraically?

Tip: When analyzing transformations of functions graphically, always consider how each transformation (stretching, compressing, reflecting) affects the coordinates of key points on the graph.