To determine the domain and range of the function shown in the graph, let's analyze the key features:

Domain:

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

• The graph shows that the function extends from the left side at $x = -\infty$ (approaching negative infinity).
• The graph stops at $x = 2$, and there is an open circle at $x = 2$, which indicates that the function is not defined at $x = 2$.

Thus, the domain of the function is $(-\infty, 2)$.

Range:

The range is the set of all possible y-values (output values) for which the function is defined.

• The graph covers values of y from positive values near 4 and goes downward.
• The lowest point on the graph occurs at $y = -5$ with a solid point at $(3, -5)$, indicating that $y = -5$ is included.
• The open circle at $(2, -2)$ indicates that $y = -2$ is not included.

Thus, the range of the function is $(-\infty, -2) \cup \{-5\}$.

Final Answer:

• Domain: $(-\infty, 2)$
• Range: $(-\infty, -2) \cup \{-5\}$

Would you like more details or further explanation on any aspect of this problem?

Here are some related questions for further practice:

1. How do you determine if a function is continuous based on its graph?
2. What is the significance of open and closed circles on a graph in terms of domain and range?
3. Can a function have the same y-value for different x-values in its domain? If so, give an example.
4. How does the concept of domain and range apply to real-world situations?
5. What changes would occur to the domain and range if the open circle at $(2, -2)$ was filled in?

Tip: Always pay attention to open and closed circles on a graph; they indicate whether a point is included in the domain or range.