To determine the domain and range of the given function based on the graph, let's analyze it carefully:

Domain:

• The domain refers to the set of all possible $x$-values (inputs) for which the function is defined.
• Observing the graph, the function starts at $x = -2$ (where the blue dot is filled) and continues until $x = 3$ (where there is an open circle).
• The closed dot at $x = -2$ indicates that the point is included, while the open circle at $x = 3$ indicates that the point is not included.

Thus, the domain is: $[-2, 3)$ (meaning the function is defined for $x$ from $-2$ to 3, but not including $x = 3$).

Range:

• The range refers to the set of all possible $y$-values (outputs) that the function can take.
• From the graph, the highest point occurs at $y = 4$, and the lowest point occurs just above $y = -5$, but since the open circle indicates it does not reach $y = -5$, the lowest point is just slightly above $-5$.
• Therefore, the range extends from $y = -5$ (not inclusive) to $y = 4$ (inclusive).

Thus, the range is: $(-5, 4]$ (meaning the function can take values from just above $-5$ up to 4, including 4).

Summary:

• Domain: $[-2, 3)$
• Range: $(-5, 4]$

Would you like further details or explanations on any part of this problem?

Here are 5 related questions to deepen your understanding:

1. What happens to the domain and range if we extend the graph further on both sides?
2. How do open and closed circles affect the domain and range of a function?
3. Can a function have a domain or range that is not continuous?
4. How would you describe the behavior of the function as $x$ approaches 3?
5. What are the implications of a function having a restricted domain in real-world scenarios?

Tip: Always remember that a filled dot means the point is included in the domain or range, while an open dot means the point is excluded.