https://file.aiquickdraw.com/mathbot/file/1725935928222a09mgf2r.jpg

The provided image defines a piecewise function \( f(x) \) as follows:

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

### Steps to sketch the graph of \( f(x) \):

1. **For \( x \leq -2 \):** 
   - The function is constant: \( f(x) = 1 \).
   - Draw a horizontal line at \( y = 1 \) for \( x \leq -2 \).
   - Since the function is defined as \( f(x) = 1 \) at \( x = -2 \), this point is included, so place a solid dot at \( (-2, 1) \).

2. **For \( -2 < x \leq 2 \):**
   - The function is \( f(x) = -x - 1 \), which is a linear equation.
   - This line has a slope of \( -1 \) and a y-intercept at \( -1 \).
   - For example, at \( x = -2 \), \( f(-2) = -(-2) - 1 = 1 \), which connects to the previous interval.
   - At \( x = 0 \), \( f(0) = -(0) - 1 = -1 \).
   - At \( x = 2 \), \( f(2) = -(2) - 1 = -3 \).
   - Since the function includes the point \( (2, -3) \), place a solid dot at \( (2, -3) \).

3. **For \( x > 2 \):**
   - The function is \( f(x) = 0 \), which is constant for all \( x > 2 \).
   - Draw a horizontal line at \( y = 0 \) for \( x > 2 \).
   - Since the function does not include \( x = 2 \), this should be an open circle at \( (2, 0) \).

### Summary of Key Points:
- A horizontal line at \( y = 1 \) for \( x \leq -2 \) with a solid point at \( (-2, 1) \).
- A linear function \( f(x) = -x - 1 \) between \( -2 < x \leq 2 \).
- A horizontal line at \( y = 0 \) for \( x > 2 \) with an open circle at \( (2, 0) \).

Would you like to see a detailed plot of this function?

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Here are 5 questions to consider based on this function:

1. How does the slope of the linear part \( f(x) = -x - 1 \) affect the steepness of the line?
2. What happens to the graph if we change the constant values in the piecewise function?
3. How does the behavior of the function change at \( x = -2 \) and \( x = 2 \)?
4. What are the domain and range of the piecewise function \( f(x) \)?
5. How would this graph change if we flipped the sign of the function for \( x > 2 \)?

**Tip:** When graphing piecewise functions, always carefully check if the boundary points are included or excluded, which affects whether the graph has open or closed dots.

Sketch a graph of f(x) where:
f(x) =
  1 if x ≤ -2,
  -x - 1 if -2 < x ≤ 2,
  0 if x > 2

This problem involves graphing a piecewise function with three parts: a constant function for x ≤ -2, a linear function for -2 < x ≤ 2, and another constant function for x > 2. Explore how to connect different parts of the graph and understand the key points at x = -2 and x = 2. Suitable for high school students in Grades 10-12.

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