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To determine which function matches \( g(x) \) based on the graph provided:

1. **Identify the transformation**: The graph of \( g(x) \) is a transformation of the graph of \( f(x) \).

2. **Analyze the transformations**:
   - **Horizontal Shift**: If \( g(x) = f(x - a) \), the graph of \( g(x) \) would be a horizontal shift of \( f(x) \) to the right by \( a \) units.
   - **Vertical Stretch**: If \( g(x) = 2f(x) \), the graph of \( g(x) \) would be a vertical stretch of \( f(x) \) by a factor of 2.

   By analyzing the graph:
   - **Horizontal Shift**: The graph of \( g(x) \) appears to be shifted to the right by 2 units compared to \( f(x) \).
   - **Vertical Stretch**: The graph of \( g(x) \) appears to be a vertical stretch of \( f(x) \) by a factor of 2.

3. **Match the transformations**:
   - The correct transformation that matches both a shift to the right by 2 units and a vertical stretch by a factor of 2 is option **D: \( g(x) = 2f(x - 2) \)**.

Thus, the correct answer is **D**.

Would you like more details or have any further questions?

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Here are five related questions for further exploration:

1. How does a horizontal shift affect the equation of a function?
2. What effect does a vertical stretch have on the graph of a function?
3. Can you explain the difference between horizontal and vertical transformations in functions?
4. How would the equation change if \( g(x) \) was a reflection of \( f(x) \)?
5. What is the effect of a vertical shift on a function's graph?

**Tip**: When dealing with transformations of functions, always pay close attention to the direction and magnitude of shifts and stretches, as they directly affect the function's equation.

Which function matches g? (based on the provided graph)

Explore how to determine which function g(x) matches based on transformations of f(x) using the given graph. This problem covers core concepts of function transformations, including horizontal shifts and vertical stretches. Suitable for students in Grades 9-11.

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