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The problem asks to find the domain and range of the transformed function \( g(x) = 2f(x + 7) - 7 \), given that the domain of \( f(x) \) is \( [-1, 17] \) and the range is \( [-4, 8] \).

### Step 1: Finding the Domain of \( g(x) \)

- The function \( g(x) = 2f(x + 7) - 7 \) applies a horizontal shift of \( x \) by -7 (due to the \( x + 7 \)).
- Shifting the domain \( [-1, 17] \) by -7 gives:
  \[
  [-1 - 7, 17 - 7] = [-8, 10]
  \]
So, the domain of \( g(x) \) is \( [-8, 10] \).

### Step 2: Finding the Range of \( g(x) \)

- The transformation includes a vertical stretch by 2 and a downward shift of 7.
- First, multiply the range of \( f(x) = [-4, 8] \) by 2:
  \[
  2 \cdot [-4, 8] = [-8, 16]
  \]
- Then, shift the result downward by 7:
  \[
  [-8, 16] - 7 = [-8 - 7, 16 - 7] = [-15, 9]
  \]
So, the range of \( g(x) \) is \( [-15, 9] \).

### Final Answer:
- Domain: \( [-8, 10] \)
- Range: \( [-15, 9] \)

Would you like more details or have any questions? 

Here are 5 related questions to expand on this:
1. How would the transformations change if the function were \( g(x) = f(x - 5) + 3 \)?
2. What happens to the range of a function if you apply a vertical stretch by 0.5 instead of 2?
3. How does a horizontal compression affect the domain of a function?
4. What is the effect of a vertical reflection on the range of a function?
5. How would adding a constant inside the function argument affect its domain?

**Tip:** When transforming functions, handle horizontal shifts (affecting the domain) separately from vertical shifts and stretches (affecting the range).

The domain of the function f(x) is [-1, 17] and the range is [-4, 8]. Using interval notation, find the domain and range of g(x) = 2f(x + 7) - 7.

Learn how to determine the domain and range of the transformed function g(x) = 2f(x + 7) - 7, where the original function f(x) has a domain of [-1, 17] and a range of [-4, 8]. This solution covers key transformation concepts like horizontal shifts and vertical stretching. Suitable for students in Grades 9-12.

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