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: \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.