To find the vertex of $w^{-1}(x)$, we first need to determine the vertex of the original function $w(x) = -\frac{2}{5} |x + 3| + 2$.

Step 1: Identify the vertex of the original function $w(x)$

The function is in the form of $w(x) = a|x - h| + k$, where the vertex is at the point $(h, k)$.

In this case:

• $a = -\frac{2}{5}$
• $|x + 3|$ means the vertex is at $x = -3$ (opposite sign inside the absolute value).
• $k = 2$

Thus, the vertex of $w(x)$ is $(-3, 2)$.

Step 2: Find the vertex of the inverse function $w^{-1}(x)$

For the inverse function $w^{-1}(x)$, the vertex will be the reverse of the vertex coordinates of $w(x)$. Therefore, the vertex of $w^{-1}(x)$ is $(2, -3)$.

Final Answer:

The vertex of $w^{-1}(x)$ is $\boxed{(2, -3)}$, which corresponds to option D.

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Would you like more details on any part of the process? Here are 5 related questions to deepen your understanding:

1. How do we find the inverse of an absolute value function like $w(x)$?
2. Why does the vertex of the inverse function swap the coordinates?
3. Can the graph of $w(x)$ help visualize the inverse function?
4. What is the significance of the sign of $a$ in determining the function's shape?
5. How can transformations like shifting and stretching affect the vertex?

Tip: When dealing with inverse functions, remember that the roles of $x$ and $y$ are reversed.