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The problem asks to determine which of the given equations defines \( g(x) \) in terms of \( f(x) \) based on the graphs.

Let's analyze the graphs:

1. **Transformation between \( f(x) \) and \( g(x) \)**:
    - The graph of \( g(x) \) appears to be a vertical stretch and a horizontal stretch (or compression) of the graph of \( f(x) \).
    - Additionally, there might be a vertical shift involved.

2. **Possible Transformations**:
    - **Vertical Scaling**: The graph of \( g(x) \) is stretched vertically compared to \( f(x) \). This suggests that \( g(x) \) could be some multiple of \( f(x) \).
    - **Horizontal Scaling**: The graph of \( g(x) \) seems horizontally compressed compared to \( f(x) \), implying that \( f(x) \) might be evaluated at a multiple of \( x \).
    - **Vertical Shift**: There could be a shift downwards, as \( g(x) \) seems to be shifted slightly compared to \( f(x) \).

Now, let's analyze the given options:

- **Option A: \( g(x) = \frac{5}{2} f(2x) \)**
    - This suggests a vertical stretch by \( \frac{5}{2} \) and a horizontal compression by \( 2 \) (as multiplying \( x \) by \( 2 \) compresses the graph horizontally).
  
- **Option B: \( g(x) = \frac{5}{2} f\left(\frac{1}{2}x\right) \)**
    - This implies a vertical stretch by \( \frac{5}{2} \) and a horizontal stretch by a factor of \( 2 \) (as multiplying \( x \) by \( \frac{1}{2} \) stretches the graph horizontally).

- **Option C: \( g(x) = 3f(2x) - 1 \)**
    - This involves a vertical stretch by \( 3 \), a horizontal compression by \( 2 \), and a vertical shift downwards by \( 1 \).

- **Option D: \( g(x) = 3 f\left(\frac{1}{2}x\right) - 1 \)**
    - This involves a vertical stretch by \( 3 \), a horizontal stretch by \( 2 \), and a vertical shift downwards by \( 1 \).

### Conclusion:

The correct answer is **C: \( g(x) = 3f(2x) - 1 \)**, because:
- The graph of \( g(x) \) is vertically stretched by a factor of 3 (compared to \( f(x) \)),
- Horizontally compressed by a factor of 2,
- And shifted down by 1 unit.

Would you like more details on any specific transformation or have any questions?

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### Related Questions:
1. How do horizontal stretches and compressions affect the graph of a function?
2. What is the effect of vertical translations on a graph?
3. Why does multiplying \( f(x) \) by a constant stretch or compress it vertically?
4. How can we verify a horizontal compression from the graph?
5. What transformations combine to give the effect seen in the graph of \( g(x) \)?

### Tip:
Always consider both horizontal and vertical changes when analyzing transformations between two functions' graphs!

The graphs of the functions f and g are given. Which of the following defines g in terms of f?

Analyze how vertical stretches, horizontal compressions, and vertical shifts affect the graph of a function. In this problem, we determine that g(x) = 3f(2x) - 1, making it suitable for Grades 10-12 students.

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