To solve for $h(x)$ based on the transformations applied to $g(x) = 2(3)^{x - 4}$:

1. Vertical Stretch by a Factor of 2: The original function $g(x) = 2(3)^{x - 4}$ will be stretched vertically by multiplying the entire function by 2. So, the new base function after the vertical stretch becomes: $h(x) = 2 \cdot 2(3)^{x - 4} = 4(3)^{x - 4}$

2. Vertical Shift Upward by 9 Units: A vertical shift upward by 9 units involves adding 9 to the entire function: $h(x) = 4(3)^{x - 4} + 9$

3. Horizontal Shift to the Right by 7 Units: To shift the function to the right by 7 units, we replace $x$ with $x - 7$: $h(x) = 4(3)^{(x - 7) - 4} + 9$ Simplifying inside the exponent: $h(x) = 4(3)^{x - 11} + 9$

Now, we compare this result with the answer choices:

• The correct answer is not listed in the provided choices directly, as none exactly match $h(x) = 4(3)^{x - 11} + 9$.

Please verify the answer choices or check if there was an omission. Let me know if you'd like more details.

Would you like more information on transformations in exponential functions?

1. How does a horizontal shift affect the graph of an exponential function?
2. Why does a vertical stretch multiply the entire function?
3. What happens when you apply both horizontal and vertical transformations?
4. How do transformations affect the domain and range of exponential functions?
5. How would a vertical compression differ from a vertical stretch?

Tip: When working with transformations, always apply multiplication factors before addition or shifts.