Let 𝑤𝑤(𝑥𝑥) = − 2
5 |𝑥𝑥 + 3| + 2 restricted over {𝑥𝑥 ≥ ℎ} where (ℎ, 𝑘𝑘)
are the coordinates of the vertex.
What is the vertex of 𝑤𝑤 −1(𝑥𝑥)?
A. (−3, 2)
B. (3, −2)
C. (−2, 3)
D. (2, −3)


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.

Let 𝑤(𝑥) = − 2/5 |𝑥 + 3| + 2 restricted over {𝑥 ≥ ℎ} where (ℎ, 𝑘) are the coordinates of the vertex. What is the vertex of 𝑤−1(𝑥)? A. (−3, 2) B. (3, −2) C. (−2, 3) D. (2, −3)

Learn how to determine the vertex of the inverse of an absolute value function w(x) = − 2/5 |x + 3| + 2. This detailed explanation includes steps to identify the vertex of the original function and swap coordinates for the inverse. Suitable for high school students studying advanced algebra.

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