https://file.aiquickdraw.com/mathbot/file/1725933985048xss5ynsk.jpg

The given piecewise function is defined as:

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

### Explanation and Graphing:

1. **For \( x \leq -1 \):**
   - The function is constant and equal to -1.
   - So, the graph is a horizontal line at \( y = -1 \) for \( x \leq -1 \).

2. **For \( -1 < x \leq 1 \):**
   - The function is linear, \( f(x) = 2x + 1 \), which has a slope of 2 and a y-intercept of 1.
   - The graph of this part is a straight line between \( x = -1 \) and \( x = 1 \).
   - At \( x = -1 \), the value is \( f(-1) = 2(-1) + 1 = -1 \), but since \( x = -1 \) is not included (strict inequality), the point \((-1, -1)\) is an open circle.
   - At \( x = 1 \), the value is \( f(1) = 2(1) + 1 = 3 \), and this point is included (closed circle).

3. **For \( x > 1 \):**
   - The function is constant and equal to 2.
   - So, the graph is a horizontal line at \( y = 2 \) for \( x > 1 \).

### Sketch:
- **For \( x \leq -1 \):** A horizontal line at \( y = -1 \).
- **From \( -1 < x \leq 1 \):** A line starting from an open circle at \((-1, -1)\), rising to a closed circle at \((1, 3)\).
- **For \( x > 1 \):** A horizontal line starting from \( (1, 2) \) as an open circle and continuing rightward at \( y = 2 \).

Would you like me to generate an image for the graph or provide more details? 

### Questions:
1. How would you calculate the slope of the linear part of the function?
2. What are open and closed circles on the graph, and why are they used here?
3. What happens to the graph when the piecewise function is constant for certain intervals?
4. How would the graph change if the condition \( x > 1 \) were instead \( x \geq 1 \)?
5. How can you determine the continuity of a piecewise function based on its graph?

### Tip:
Always check for continuity and smooth transitions at the boundaries between intervals when graphing piecewise functions.

Sketch a graph of the piecewise function: f(x) = {-1 if x ≤ -1; 2x + 1 if -1 < x ≤ 1; 2 if x > 1}.

This problem involves graphing a piecewise function with three segments: a constant segment for x ≤ -1, a linear segment for -1 < x ≤ 1, and another constant segment for x > 1. Learn the step-by-step approach to sketching the graph, including handling open and closed circles at the endpoints.

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