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The problem involves the transformation of the function \( y = f(x) \), which has a domain of \( -4 \leq x \leq 6 \), to the function \( y = h(x) = -3f(2x) - 4 \). We need to find the domain of \( h(x) \) based on the given transformation.

### Steps:
1. The original domain of \( f(x) \) is \( -4 \leq x \leq 6 \).
2. The transformation \( h(x) = -3f(2x) - 4 \) involves:
   - A horizontal scaling by a factor of \( 2 \) (due to the \( 2x \)).
   
   The factor \( 2 \) affects the input \( x \) such that:
   \[
   f(2x) \quad \text{means} \quad -4 \leq 2x \leq 6
   \]
3. To solve for \( x \), divide the entire inequality by \( 2 \):
   \[
   -2 \leq x \leq 3
   \]
4. This implies that the domain of \( h(x) \) is \( -2 \leq x \leq 3 \).

Thus, the correct domain for \( h(x) \) is \( -2 \leq x \leq 3 \), which corresponds to the last answer option.

Would you like further explanation on how transformations affect the domain?

Here are 5 related questions to consider:
1. How does vertical scaling affect the range of a function?
2. What happens to the domain if we shift \( f(x) \) horizontally instead of scaling it?
3. How would the range of \( h(x) \) be affected by the transformation?
4. How do multiple transformations (scaling and translation) affect a function's graph?
5. What role does the factor of \( -3 \) play in transforming the function?

**Tip:** When dealing with transformations, always consider how scaling, reflections, and shifts affect both the domain and range of the original function.

The function y = f(x) has domain -4 ≤ x ≤ 6 and range 0 ≤ x ≤ 10. The graph of y = h(x) is the result of the transformation h(x) = -3f(2x) - 4. Which of the following gives the domain of h(x)?

Learn how to find the domain of a transformed function h(x) = -3f(2x) - 4. This problem covers function transformations involving horizontal scaling and translations, suitable for high school algebra students. The original domain is -4 ≤ x ≤ 6, and the transformed domain is -2 ≤ x ≤ 3.

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