https://file.aiquickdraw.com/mathbot/file/1728271545548mh3jj76c.jpg

The given image shows a piecewise function and its graph. The function \( f(x) \) is defined as:

\[
f(x) =
\begin{cases} 
-3 & \text{for } x \leq -3 \\
2x - 1 & \text{for } -3 < x < 1 \\
-x - 1 & \text{for } 1 \leq x
\end{cases}
\]

### Analyzing the function and the graph:
1. **For \( x \leq -3 \)**: The function is constant, \( f(x) = -3 \). This is represented by a horizontal line at \( y = -3 \) for all \( x \leq -3 \).
   
2. **For \( -3 < x < 1 \)**: The function is linear, \( f(x) = 2x - 1 \). This is a line with slope 2 and y-intercept -1. In this interval, the graph shows a segment starting at \( x = -3 \) and ending at \( x = 1 \). The value of the function at \( x = -3 \) is \( f(-3) = 2(-3) - 1 = -7 \), and the value at \( x = 1 \) is \( f(1) = 2(1) - 1 = 1 \).
   
   - The graph correctly shows an open circle at \( x = 1 \), meaning the function is not defined at \( x = 1 \) for this piece.

3. **For \( x \geq 1 \)**: The function is again linear, \( f(x) = -x - 1 \), a line with slope -1 and y-intercept -1. This part starts at \( x = 1 \). At \( x = 1 \), \( f(1) = -1 - 1 = -2 \), which the graph should show as a filled circle at \( (1, -2) \).

### The graph:

The graph appears to be mostly correct:
- There is a horizontal line at \( y = -3 \) for \( x \leq -3 \).
- A linear segment for \( -3 < x < 1 \) with slope 2, which ends in an open circle at \( x = 1 \).
- A linear segment for \( x \geq 1 \) with slope -1, starting at \( (1, -2) \).

Would you like further details or clarifications? Here are some related questions for deeper understanding:

1. How do we derive the slopes of each line in the piecewise function?
2. How does the open circle at \( x = 1 \) relate to continuity in this function?
3. What is the significance of the slope in each interval of the piecewise function?
4. Can you describe how the graph behaves at the transition points \( x = -3 \) and \( x = 1 \)?
5. How does the graph change if one of the conditions for the function is altered?

**Tip**: When graphing piecewise functions, always check for open and closed circles at transition points to ensure the correct domain boundaries are represented.

Graph the following piecewise function: f(x) = -3 when x <= -3, 2x - 1 when -3 < x < 1, -x - 1 when 1 <= x

Explore how to graph a piecewise function with linear equations in each interval. This problem demonstrates graphing techniques for piecewise functions, including recognizing open and closed circles at transition points. Suitable for Grades 9-12.

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