To determine the domain and range of the given piecewise function from the graph, let's analyze the x-values and y-values where the function is defined.

Domain:

The domain of a function consists of all the x-values for which the function is defined. Let's look at the graph:

1. The function starts at $x = -4$ and continues to $x = -2$.
2. There is a gap between $x = -2$ and $x = 0$.
3. The function is defined again from $x = 0$ to $x = 3$.

• From the graph:
  • The function exists from $x = -4$ to $x = -2$ (both endpoints are included, indicated by filled circles).
  • Then from $x = 0$ to $x = 3$ (both endpoints are included).

Thus, the domain is: [ [-4, -2] \cup [0, 3] ]

Range:

The range of a function consists of all the y-values the function takes.

1. For the interval from $x = -4$ to $x = -2$:

   • The y-values range from $y = 1$ to $y = -3$ (since there is a continuous line from point $(-4, 1)$ to $(-2, -3)$).

2. For the interval from $x = 0$ to $x = 3$:

   • The y-values range from $y = -2$ to $y = 2$ (from point $(0, -2)$ to $(3, 2)$).

• Combining these intervals:
  • The range from $y = -3$ to $y = 1$ (for the first segment).
  • The range from $y = -2$ to $y = 2$ (for the second segment).

Thus, the range is: [ [-3, 1] \cup [-2, 2] ]

Final Answer:

• Domain: [ [-4, -2] \cup [0, 3] ]
• Range: [ [-3, 1] \cup [-2, 2] ]

Would you like a more detailed explanation or have any other questions?

Additional Questions to Explore:

1. What is a piecewise function, and how is it defined mathematically?
2. How do you determine if an endpoint is included or excluded from a function's domain or range?
3. Can a function have multiple disjointed intervals for its domain or range?
4. What are some real-world examples where piecewise functions are applied?
5. How does one determine the continuity of a piecewise function?

Tip: Always pay attention to the type of circles (filled or open) on a graph, as they indicate whether the endpoints are included or excluded.