https://file.aiquickdraw.com/mathbot/file/1725666585484a0ktgjuz.jpg

The piecewise function provided is:

\[
g(x) = 
\begin{cases} 
x, & x \leq -3 \\
-x + 3, & -3 < x < 2 \\
5, & x > 2
\end{cases}
\]

### Let's break down each part of the function:

1. **For \(x \leq -3\):** The function is \(g(x) = x\), which is a linear equation with a slope of 1. The graph will be a line that follows \(y = x\) for all \(x \leq -3\).

2. **For \(-3 < x < 2\):** The function is \(g(x) = -x + 3\), another linear function but with a slope of \(-1\) and a y-intercept at \(3\). This line only exists in the interval \(-3 < x < 2\).

3. **For \(x > 2\):** The function is \(g(x) = 5\), which is a horizontal line at \(y = 5\) for all \(x > 2\).

### Graphical behavior:

- The first part is a straight line starting from \(x = -3\) and moving leftwards with a slope of 1.
- The second part is a line segment with a slope of \(-1\), starting after \(x = -3\) and stopping at \(x = 2\).
- The third part is a horizontal line at \(y = 5\), starting from \(x = 2\).

Do you want a detailed step-by-step construction of the graph? Here are some questions to consider for better understanding:

1. How would you calculate specific points for each interval of the function?
2. What would happen if we change the conditions of the piecewise function?
3. What are the key points (like where the function changes behavior) in this function?
4. How do you find the continuity of this piecewise function?
5. How do slopes of lines affect the shape of the graph in piecewise functions?

**Tip:** When graphing piecewise functions, always pay close attention to the boundaries (like \(x = -3\) and \(x = 2\)) and whether the endpoints are included or excluded (denoted by open or closed circles).

Match the piecewise function with its graph: 
 g(x) = { x, if x ≤ -3; -x + 3, if -3 < x < 2; 5, if x > 2 }

Learn how to match a piecewise function with its graph. This problem involves three parts: g(x) = x for x ≤ -3, g(x) = -x + 3 for -3 < x < 2, and g(x) = 5 for x > 2. Explore how each segment of the function behaves and graph it accurately. Suitable for Grades 9-12.

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