https://file.aiquickdraw.com/mathbot/file/1726286433349g71z3z8o.jpg

To analyze this piecewise graph, we need to determine the **domain** and **range** of the function.

### Domain:
The domain refers to all the x-values for which the function is defined. From the graph, it seems that the function is composed of three distinct segments:
1. The first segment is a line increasing to the right, starting from approximately \(x = -9\) and stopping at \(x = -2\) with an open circle at \(x = -2\), meaning the function is not defined at \(x = -2\).
2. The second segment is a parabola-like curve starting from \(x = -2\) and ending at \(x = 1\). The function is not defined at \(x = -2\) (open circle), but it is defined at \(x = 1\) (closed circle).
3. The third segment is a horizontal line starting at \(x = 1\) and continuing to the right, beyond the edge of the graph.

Thus, the domain is:
\[
\text{Domain} = (-9, -2) \cup (-2, 1] \cup [1, \infty)
\]
It means that the function is not defined at \(x = -2\), but it is defined for all other points in the intervals.

### Range:
The range refers to all possible y-values of the function. Observing the graph:
1. The first segment covers y-values from around \(y = 0\) to \(y \approx 4.5\).
2. The second segment (the parabola-like curve) reaches a minimum value at \(y = -4\) and a maximum at \(y = 4\).
3. The third segment is a horizontal line at \(y = 2\).

Thus, the range is:
\[
\text{Range} = [-4, 4] \cup \{2\}
\]
This means that the function takes on all y-values between \(-4\) and 4 (including these values), and specifically, it takes \(y = 2\) due to the third segment.

### Summary:
- **Domain**: \( (-9, -2) \cup (-2, 1] \cup [1, \infty) \)
- **Range**: \( [-4, 4] \cup \{2\} \)

Would you like to go into more detail or have any questions?

### Related Questions:
1. How do we formally define piecewise functions?
2. What is the significance of open and closed circles in graphs?
3. How does one interpret horizontal and vertical asymptotes in graphs?
4. How do we algebraically find the domain and range of more complex functions?
5. Can the range of a function ever be infinite in both directions?

### Tip:
When determining the domain and range, always pay attention to open and closed circles, as they indicate whether the function includes or excludes those particular values at given points.

Determina el dominio y rango para cada una de las gráficas de funciones seccionadas dadas

Analyze a piecewise function graph to determine its domain and range. The function consists of multiple segments including a linear function, a parabola, and a horizontal line. This problem helps students in Grades 9-11 understand how to read graphs and interpret domain and range.

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