Let's analyze the problem step by step.

The given function $f(x)$ has:

• Domain: $[-4, 2]$
• Range: $[-2, 6]$

When we form the new function $h(x) = f(x + 5)$, the input shift by $+5$ affects the domain but does not alter the range. Here's why:

1. Domain of $h(x)$:

   • Since $h(x) = f(x+5)$, the input $x+5$ must lie in the domain of $f(x)$, which is $[-4, 2]$.
   • Solving for $x$, we subtract 5 from each side: $-4 \leq x+5 \leq 2 \quad \implies \quad -9 \leq x \leq -3$
   • Therefore, the domain of $h(x)$ is $[-9, -3]$.

2. Range of $h(x)$:

   • The range of $h(x)$ is the same as the range of $f(x)$, since shifting $x$ horizontally does not affect the vertical output.
   • Thus, the range of $h(x)$ is $[-2, 6]$.

Correct Option:

The correct choice is: Domain of $h$: $[-9, -3]$; Range of $h$: $[-2, 6]$.

Would you like a detailed explanation of any part? Here are related questions to expand understanding:

1. What happens to the domain and range if $x+5$ becomes $x-5$?
2. How does horizontal shifting affect the graph of a function in general?
3. Can the range of a function change under transformations, and why?
4. How would the domain and range change if $h(x) = 2f(x+5)$?
5. Can we determine the new graph of $h(x)$ without calculating points explicitly?

Tip: Shifting a function horizontally by $x+c$ adjusts the domain, but the range stays the same unless there are additional vertical transformations.