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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.

Explore how to determine the correct equation for g(x) in terms of f(x) by analyzing transformations from the graph of y = f(x) to y = g(x). This problem involves understanding vertical and horizontal transformations, reflections, and compressions in function graphs. Suitable for Grades 10-12.

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