Let's break down how to determine the domain and range of $g(x) = f(x + 8)$.

Step 1: Determine the Domain of $g(x)$

The domain of $f(x)$ is given as $[-15, -4]$. For the function $g(x) = f(x + 8)$, we perform a shift in the $x$-values. Specifically, the argument of $f(x)$ is $x + 8$, which means the function $f$ is evaluated at $x + 8$ instead of $x$.

To maintain the validity of $f(x)$, the expression $x + 8$ must fall within the domain of $f(x)$, which is $[-15, -4]$.

Thus, we need to solve for $x$ such that: $-15 \leq x + 8 \leq -4$ Subtract 8 from all parts of the inequality: $-15 - 8 \leq x \leq -4 - 8$ $-23 \leq x \leq -12$ Therefore, the domain of $g(x)$ is $[-23, -12]$.

Step 2: Determine the Range of $g(x)$

The range of $f(x)$ is given as $[5, 11]$. Since $g(x) = f(x + 8)$ is just a horizontal shift of $f(x)$, the range of $g(x)$ will remain the same as the range of $f(x)$.

Thus, the range of $g(x)$ is also $[5, 11]$.

Final Answer:

• The domain of $g(x)$ is $\boxed{[-23, -12]}$.
• The range of $g(x)$ is $\boxed{[5, 11]}$.

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Would you like further explanation or have any questions about this?

Here are 5 related questions to explore further:

1. How does a horizontal shift affect the domain and range of a function?
2. What happens if the function $f(x)$ is shifted vertically instead of horizontally?
3. How can we find the domain and range of composite functions like $g(x) = f(x + c)$?
4. How do shifts in functions affect the graph in terms of transformations?
5. How would the domain and range change if we used $g(x) = f(x - 8)$?

Tip: When shifting a function horizontally, always adjust the domain by shifting the interval for $x$, while the range stays unchanged.