To find the domain and range of the function $g(x) = f(x+7)$, let’s analyze the transformations.

Step 1: Analyze the Domain

The domain of the original function $f(x)$ is $(-\infty, \infty)$.

The transformation $g(x) = f(x+7)$ involves a horizontal shift of the graph of $f(x)$ by 7 units to the left. A horizontal shift does not change the domain, so the domain of $g(x)$ remains the same as $f(x)$:
$\text{Domain of } g(x): (-\infty, \infty)$

Step 2: Analyze the Range

The range of $f(x)$ is $[-15, 0]$. Horizontal shifts do not affect the range, so the range of $f(x+7)$ is still $[-15, 0]$.

Final Answer

$\text{Domain of } g(x): (-\infty, \infty)$
$\text{Range of } g(x): [-15, 0]$

Let me know if you need detailed explanations or have questions!

Related Questions:

1. How does a vertical shift affect the range of a function?
2. Can the domain of a function ever be reduced by a horizontal shift?
3. What would the range of $g(x) = f(x+7) - 3$ be in this scenario?
4. If $f(x)$ were restricted to $x > 0$, how would this change the domain of $g(x)$?
5. Why do horizontal shifts not affect the range of a function?

Tip: Remember that horizontal transformations affect only the domain, while vertical transformations affect only the range.