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To write the function \( g(x) \) based on the transformation of the graph of \( f(x) \), we need to analyze how the graph of \( f \) has been transformed to create the graph of \( g \).

### Step-by-step analysis:
1. **Original function \( f(x) \)**: The function \( f(x) \) appears to be a V-shaped graph, which is typical of the absolute value function. Based on the image, the graph of \( f(x) \) seems to be:
   \[
   f(x) = |x|
   \]
   since the graph has a vertex at the origin \((0, 0)\) and slopes of 1 and -1.

2. **Transformation of \( f(x) \) to \( g(x) \)**:
   - The graph of \( g(x) \) is the result of reflecting \( f(x) \) across the x-axis, as the V-shape is inverted.
   - Additionally, the graph of \( g(x) \) is **shifted downward by 2 units**.

### Conclusion:
The transformation can be described as a vertical reflection and a downward shift. This transformation can be written as:
   \[
   g(x) = -f(x) - 2
   \]

Since \( f(x) = |x| \), substituting this in gives:
   \[
   g(x) = -|x| - 2
   \]

Thus, the function \( g(x) \) is:
   \[
   g(x) = -|x| - 2
   \]

Would you like further clarification or details on this solution?

Here are five related questions to explore:
1. What are other common transformations of absolute value functions?
2. How does a horizontal shift affect the equation of an absolute value function?
3. What is the general formula for reflecting a function across the x-axis?
4. How does vertical stretching or compression affect the graph of a function?
5. Can you provide an example of a horizontal reflection and how that changes the function?

**Tip:** Always pay attention to whether a transformation involves shifts, reflections, or stretches/compressions to properly adjust the function's equation.

Write a function g whose graph represents the indicated transformation of the graph of f.

This problem involves the transformation of an absolute value function, including a vertical reflection and downward shift. The original function f(x) = |x| is transformed to g(x) = -|x| - 2. Learn how to analyze and solve such transformations step by step, suitable for Grades 9-12.

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