Given the function $g(x) = f(-(x+8)) - 6$, we are tasked with finding the domain and range of $g(x)$ based on the information about $f(x)$.

Step 1: Analyzing the Domain

The domain of $f(x)$ is $[-6, 13]$, and we want to determine how the transformation affects the domain for $g(x)$.

1. Inside the function $f$, we have $-(x+8)$: $y = -(x+8) = -x - 8$ So, $f(-(x+8))$ means that the input to $f$ is transformed by $-(x+8)$.

2. We reverse the effect to find the domain of $g(x)$. Let’s find for which $x$, $-(x+8)$ lies within the domain of $f(x)$ (which is $[-6, 13]$).

   • Start by solving for $x$ when $-(x+8) = -6$: $-(x+8) = -6 \implies x + 8 = 6 \implies x = -2$
   • Now solve for $x$ when $-(x+8) = 13$: $-(x+8) = 13 \implies x + 8 = -13 \implies x = -21$

   So, the domain of $g(x)$ is the interval $[-21, -2]$.

Step 2: Analyzing the Range

The range of $f(x)$ is $[-15, 4]$, and the transformation $f(-(x+8)) - 6$ shifts the range by $-6$.

• If $f(x)$ has a range of $[-15, 4]$, subtracting 6 from each value results in a new range of: [ [-15 - 6, 4 - 6] = [-21, -2] ]

Final Answer:

• The domain of $g(x)$ is $[-21, -2]$.
• The range of $g(x)$ is $[-21, -2]$.

Would you like further clarification or additional details on any part of this?

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Related Questions:

1. What happens to the range of $g(x)$ if we change the constant shift outside the function?
2. How does reflecting the input of $f(x)$ affect the graph of the function?
3. Can you find the inverse of $g(x) = f(-(x+8)) - 6$?
4. How does a horizontal shift impact the domain of a function in general?
5. If $f(x)$ had a different domain, how would that affect $g(x)$?

Tip: Always check how transformations like shifting, reflecting, or stretching affect both the domain and range of a function.