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We are given two functions:

- \( f(x) = |x| \)
- \( g(x) = (x - 2)^2 - 4 \)

We are asked to determine which combination of these functions is represented by the graph.

### Step 1: Analyze the graph
The graph shows a parabola that opens downwards, with its vertex at \( (2, 0) \), and another part that forms a sharp "V" shape. This suggests that the graph is likely a combination of an absolute value function (from \( f(x) = |x| \)) and a quadratic function (from \( g(x) = (x - 2)^2 - 4 \)).

### Step 2: Check function \( g(x) \)
The function \( g(x) = (x - 2)^2 - 4 \) represents a parabola that has been shifted 2 units to the right and 4 units downward. The vertex of this parabola is at \( (2, -4) \). This matches the shape of the lower part of the graph.

### Step 3: Check function \( f(x) \)
The function \( f(x) = |x| \) represents a "V"-shaped graph, where the sharp point is at \( (0, 0) \). However, the absolute value function could appear as part of a composite function if the shape of \( g(x) \) is involved.

### Step 4: Analyze the possible compositions
Let's explore the given options:

- **A.** \( y = f(g(x)) \): This would apply the absolute value function to \( g(x) \). Since \( g(x) \) reaches negative values for \( x \) outside the interval \( (0, 4) \), applying \( f(g(x)) \) would turn those negative values positive. This would match the graph, where the negative part of the parabola becomes positive, forming a "W" shape.
  
- **B.** \( y = g(f(x)) \): This would mean applying the quadratic function \( g(x) \) to the absolute value function \( f(x) \). This does not match the graph, as it would produce a different shape.
  
- **C.** \( y = f(x) \cdot g(x) \): This represents the product of the two functions, which would result in a much more complicated shape, not the one in the graph.
  
- **D.** \( y = f(x) + g(x) \): This would result in adding the two functions together, which would not produce the correct shape either.

### Conclusion:
The correct answer is **A. \( y = f(g(x)) \)**, which reflects the graph's shape, where the negative part of \( g(x) \) has been reflected upwards.

Would you like more details on any part of this solution? Here are some follow-up questions you might find interesting:

1. How does the composition \( f(g(x)) \) affect the graph compared to \( g(x) \) alone?
2. What is the significance of the absolute value function in transforming negative values?
3. Can we generalize how to predict the effect of composing functions?
4. How would \( g(f(x)) \) differ graphically from \( f(g(x)) \)?
5. What is the effect of shifting and scaling quadratic functions like \( g(x) \)?

**Tip:** When graphing composite functions, always look for where one function changes sign, as absolute value functions often cause sharp turns in the graph.

Suppose that f(x) = |x| and g(x) = (x - 2)^2 - 4. Which of the following functions does the given graph represent? (Options: A) y = f(g(x)), B) y = g(f(x)), C) y = f(x) * g(x), D) y = f(x) + g(x))

Explore how to determine the correct composite function that matches the given graph. The problem involves absolute value functions, quadratic functions, and graph transformations. Suitable for students in Grades 10-12.

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