The problem provided asks for the transformation of the function $f(x) = 2^{x+1} - 4$ to obtain $g(x)$. Here's how we approach the solution based on the series of transformations:

1. Reflection over the horizontal axis: Reflecting a function $f(x)$ over the x-axis changes the function to $-f(x)$. $f(x) = 2^{x+1} - 4 \quad \Rightarrow \quad -f(x) = -(2^{x+1} - 4) = -2^{x+1} + 4$ So, after reflection, the function becomes: $-f(x) = -2^{x+1} + 4$

2. Translation by the vector $\begin{pmatrix} 0 \\ -6 \end{pmatrix}$: Translating a function vertically by $-6$ means we subtract 6 from the function. This gives: $-f(x) - 6 = (-2^{x+1} + 4) - 6 = -2^{x+1} - 2$ Now, the function becomes: $g(x) = -2^{x+1} - 2$

3. Vertical dilation with a scale factor of $\frac{3}{2}$: Vertical dilation multiplies the entire function by the scaling factor. So, we multiply by $\frac{3}{2}$: $g(x) = \frac{3}{2}(-2^{x+1} - 2) = \frac{3}{2} \cdot (-2^{x+1}) - \frac{3}{2} \cdot 2$ Simplifying: $g(x) = -3 \cdot 2^{x+1} - 3$

Thus, the transformed function is: $g(x) = -3 \cdot 2^{x+1} - 3$

Let me know if you need further clarification or more details.

Here are 5 related questions to explore this concept:

1. How does reflecting a function over the x-axis affect its graph?
2. What happens to a function when it is translated vertically by a constant?
3. How do you apply vertical dilation to any given function?
4. What is the effect of each individual transformation (reflection, translation, dilation) on the graph?
5. Can these transformations be applied in a different order, and if so, would the result be the same?

Tip: Always apply transformations step by step, and remember that the order of transformations (especially involving translations and dilations) can significantly affect the result.