Let's analyze the graph and answer the questions about the domain and range of the piecewise-defined function.

Domain:

The domain is the set of all possible $x$-values for which the function is defined.

• The graph starts at $x = -9$ and ends at $x = 8$.
• Between these points, we notice gaps. Specifically:
  • There’s a gap between $x = -6$ and $x = -1$, where no part of the graph is present.

Therefore, the domain includes:

• From $x = -9$ to $x = -6$,
• And from $x = -1$ to $x = 8$.

Thus, the correct domain is: Option B: $[-9, -1] \cup [3, 8]$.

Range:

The range is the set of all possible $y$-values that the function attains.

• The lowest point on the graph is at $y = -6$, and the highest is at $y = 7$.
• The function covers all values between $-6$ and 7, inclusively.

Thus, the correct range is: Option C: $[-6, 7]$.

Let me know if you would like further details or explanations!

Relative Questions:

1. How do gaps in a graph affect the domain of a function?
2. What does it mean for a function to have a limited range?
3. Can a function have multiple intervals in its domain, and how do you identify them?
4. What is the significance of the endpoints on a graph in determining domain and range?
5. How can you verify that a graph includes or excludes a value for $x$ or $y$?

Tip:

When analyzing a graph for domain and range, always check for any discontinuities (gaps) in the $x$- and $y$-values to ensure you account for intervals where the function is not defined.