https://file.aiquickdraw.com/mathbot/file/172956877695078qdsw42.jpg

The image you uploaded presents a piecewise function, defined as:

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

Let's break down the function step-by-step and plot it on the provided graph:

1. **For \( x \leq -2 \):**  
   The function is constant at \( f(x) = 3 \). This means there is a horizontal line at \( y = 3 \) for all \( x \leq -2 \).

2. **For \( -2 < x \leq 1 \):**  
   The function is linear: \( f(x) = -x + 1 \), which has a slope of -1 and a y-intercept of 1. You can plot this as a straight line starting at \( (-2, 3) \) and ending at \( (1, 0) \).

3. **For \( x > 1 \):**  
   The function is constant at \( f(x) = 1 \), which is a horizontal line at \( y = 1 \) for all \( x > 1 \).

Now, let's describe what you should expect in the graph:

- A **horizontal line at \( y = 3 \)** extending leftward from \( x = -2 \) (including the point \( (-2, 3) \)).
- A **linear segment** connecting \( (-2, 3) \) to \( (1, 0) \).
- A **horizontal line at \( y = 1 \)** starting from \( x = 1 \) and extending to the right.

Would you like to know how to compute specific points or have more details?

Here are 5 related questions to expand your understanding:
1. What is the significance of piecewise functions in real-world applications?
2. How do you find the slope and intercept of linear parts of a piecewise function?
3. How can we determine if the function is continuous at \( x = -2 \) and \( x = 1 \)?
4. What is the domain and range of the piecewise function provided?
5. How does changing the slope of the linear segment affect the overall graph?

**Tip:** Always mark open or closed circles at boundary points when graphing piecewise functions to show whether the endpoints are included or excluded.

Sketch a graph of f(x) = { 3 if x ≤ -2, -x + 1 if -2 < x ≤ 1, 1 if x > 1 }

Explore how to sketch the graph of a piecewise function where f(x) = 3 for x ≤ -2, f(x) = -x + 1 for -2 < x ≤ 1, and f(x) = 1 for x > 1. This tutorial breaks down the steps for graphing each segment and checking continuity.

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