https://file.aiquickdraw.com/mathbot/file/1725420786472lkn03tx2.jpg

The graph in the image consists of two separate pieces: one is a line segment on the left, and the other is a part of a parabola on the right. Let's analyze the domain and range for the entire graph.

### Domain:
The domain is the set of all possible \( x \)-values for which the graph is defined.

- The left segment starts at \( x = -7 \) and ends at \( x = -2 \).
- The right curve starts at \( x = 1 \) and continues indefinitely upwards to the right.

So, the domain is:
\[
\text{Domain} = [-7, -2] \cup [1, \infty)
\]

### Range:
The range is the set of all possible \( y \)-values that the graph attains.

- For the left segment, the lowest \( y \)-value is \( y = -4 \) at \( x = -2 \), and the highest \( y \)-value is \( y = 1 \) at \( x = -7 \).
- For the right curve (a parabola), the lowest point is at \( y = 2 \) when \( x = 1 \), and the \( y \)-values increase indefinitely as \( x \) increases.

So, the range is:
\[
\text{Range} = [-4, 1] \cup [2, \infty)
\]

### Final Answers:
- **Domain**: \([-7, -2] \cup [1, \infty)\)
- **Range**: \([-4, 1] \cup [2, \infty)\)

Would you like further details on how these intervals were determined, or do you have any questions? Here are some related questions to expand on this concept:

1. How would the domain and range change if the right side of the graph extended horizontally instead of vertically?
2. What is the significance of open versus closed intervals in the domain and range?
3. How can you determine if a function is continuous based on its graph?
4. What happens to the domain and range if the graph includes a vertical or horizontal asymptote?
5. How would the graph look if the piecewise function included a linear section instead of a parabolic curve?

**Tip:** When determining the domain and range from a graph, always look for endpoints and any gaps or jumps in the graph to identify where the function is and isn't defined.

Determine the domain and range of the given graph.

This guide explains how to determine the domain and range of a piecewise function using a graphical approach. It covers key mathematical concepts such as domain, range, and analyzing graphs. Ideal for students in Grades 8-10.

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