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Let's go through each part of this exercise step by step:

### Given:
The function \( f(x) = 2|x| \), which is the absolute value function scaled by 2. 

From the graph:
- The blue and green lines represent the graph of \( f(x) \).
- The red line represents the line of reflection (likely \( y = x \)) across which the reflection will occur.

Now, let's address each part:

---

### (a) Label \( f \) and \( f^{-1} \). Is \( f^{-1} \) a function?

- **Graphing \( f \):** The function \( f(x) = 2|x| \) is V-shaped, with a slope of 2 on both sides of the y-axis. It is defined as:
  \[
  f(x) =
  \begin{cases}
  2x & \text{if } x \geq 0, \\
  -2x & \text{if } x < 0.
  \end{cases}
  \]
  
- **Is \( f^{-1} \) a function?** To find the inverse, we need to reflect the graph across the line \( y = x \), and we can see from the graph that the reflection does not pass the vertical line test (as there are multiple y-values for a single x-value). Therefore, **\( f^{-1} \) is not a function** because it fails to satisfy the criteria of a function (each input must map to only one output).

---

### (b) Take \( g \) to be the restriction of \( f \) to \( (-2, 0] \cup [2, \infty) \). Sketch \( g \).

- We are restricting \( f(x) = 2|x| \) to the intervals \( (-2, 0] \) and \( [2, \infty) \). On these intervals:
  - For \( (-2, 0] \), \( f(x) = -2x \).
  - For \( [2, \infty) \), \( f(x) = 2x \).

  The graph of \( g \) would consist of:
  - A line segment from \( (-2, 4) \) to \( (0, 0) \) for \( (-2, 0] \).
  - A ray starting at \( (2, 4) \) and extending infinitely upwards for \( [2, \infty) \).
  
---

### (c) Reflect \( g \) across the line \( y = x \) to obtain the function \( g^{-1} \).

- Reflecting \( g \) across the line \( y = x \):
  - The segment from \( (-2, 4) \) to \( (0, 0) \) will be reflected to a segment from \( (4, -2) \) to \( (0, 0) \).
  - The ray from \( (2, 4) \) will be reflected to a ray starting from \( (4, 2) \), extending infinitely to the right.

---

### (d) Determine \( g^{-1}(f(4)) \), \( g^{-1}(f(-1)) \), and \( g^{-1}(f(-4)) \).

- **\( f(4) = 2|4| = 8 \)**: Now, we need \( g^{-1}(8) \).
  - From the inverse of \( g \), the point \( (8, 4) \) reflects to \( (4, 8) \), so \( g^{-1}(8) = 4 \).

- **\( f(-1) = 2| -1 | = 2 \)**: Now, we need \( g^{-1}(2) \).
  - From the inverse of \( g \), the point \( (2, 4) \) reflects to \( (4, 2) \), so \( g^{-1}(2) = 2 \).

- **\( f(-4) = 2| -4 | = 8 \)**: Again, we need \( g^{-1}(8) \), which is the same as \( f(4) \), so \( g^{-1}(8) = 4 \).

---

Would you like more details on any part of this explanation? 

Here are some additional questions related to this problem:

1. What is the general approach to finding the inverse of an absolute value function?
2. How do we use the reflection technique to verify if an inverse is a function?
3. How can restricting the domain make a non-invertible function invertible?
4. What are the key differences between the original function and its inverse graphically?
5. How would you modify the graph if the function \( f(x) \) was shifted?

**Tip:** Always check the domain and range of both the function and its inverse to ensure you're working within allowable values.

Take f to be the function that is given by f(x) = 2|x|. Analyze its inverse and reflect the function across the line y=x, and perform other operations such as restricting domains and calculating inverse function values.

Explore how to analyze the function f(x) = 2|x|, find its inverse, restrict domains, and reflect the function across the line y=x. This problem covers absolute value functions, inverse functions, domain restrictions, and the reflection technique. Suitable for Grade 11-12 students in Pre-Calculus or Calculus.

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