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).