The domain of the function f(x) is open square bracket, minus, 6, comma, 13, close square bracket[−6,13] and the range is open square bracket, minus, 15, comma, 4, close square bracket[−15,4]. Using interval notation, find the domain and range of g, of, x, equals, f, of, minus, left parenthesis, x, plus, 8, right parenthesis, minus, 6g(x)=f(−(x+8))−6

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.

The domain of the function f(x) is [−6,13] and the range is [−15,4]. Using interval notation, find the domain and range of g(x) = f(−(x+8))−6.

This page provides a step-by-step solution for finding the domain and range of g(x) = f(−(x+8))−6. It covers key function transformation concepts, including shifts and reflections, suitable for students in Grades 10-12. The original function f(x) has a domain of [−6, 13] and a range of [−15, 4], with transformations altering both.

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