https://file.aiquickdraw.com/mathbot/file/1725061654796up0p1f66.jpg

To determine the domain and range of the piecewise linear function shown in the image, let's analyze the graph step by step.

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

- The graph starts at \( x = -5 \) and ends at \( x = 5 \).
- Therefore, the domain is all \( x \) values between \( -5 \) and \( 5 \), inclusive.

**Domain:**
\[
\text{Domain} = [-5, 5]
\]

### Range:
The **range** of the function is the set of all possible \( y \) values that the function can take.

- The lowest \( y \) value on the graph is \( -6 \), which occurs at both \( x = -5 \) and \( x = 5 \).
- The highest \( y \) value occurs at \( y = 4 \) when \( x \) is between \( -2 \) and \( -1 \).

**Range:**
\[
\text{Range} = [-6, 4]
\]

### Summary:
- **Domain:** \([-5, 5]\)
- **Range:** \([-6, 4]\)

Would you like any further details or have any questions?

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Here are some related questions you might be interested in:

1. What is a piecewise function, and how is it defined?
2. How do you determine the slope of each segment in the graph?
3. Can the domain and range of a function be represented in set notation?
4. How would the domain and range change if the graph were extended?
5. How do vertical and horizontal shifts affect the domain and range of a function?

**Tip:** Always pay close attention to the endpoints of the graph when determining the domain and range—they often indicate whether the intervals should be open or closed.

Explore how to determine the domain and range of piecewise linear functions through graph analysis. This detailed explanation covers the fundamental concepts of domain and range in piecewise functions, providing step-by-step insights into identifying intervals and endpoints on graphs. Suitable for students in Grades 9-12.

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