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The function \( f(x) \) is defined piecewise as follows:

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

### To sketch the graph of \( f(x) \):

1. **For \( x \leq -1 \):**
   - \( f(x) = 0 \), which is a constant function. This corresponds to a horizontal line at \( y = 0 \) for all values of \( x \leq -1 \).
   - The point \( (-1, 0) \) is included in this part of the function.

2. **For \( -1 < x \leq 2 \):**
   - \( f(x) = -x - 1 \), which is a linear function with slope \( -1 \) and y-intercept \( -1 \). The graph will be a straight line between \( x = -1 \) and \( x = 2 \).
   - At \( x = -1 \), the function gives \( f(-1) = -(-1) - 1 = 0 \), which matches the boundary point.
   - At \( x = 2 \), the function gives \( f(2) = -(2) - 1 = -3 \), so the point \( (2, -3) \) is part of the graph.

3. **For \( x > 2 \):**
   - \( f(x) = -2 \), another constant function. This is a horizontal line at \( y = -2 \) for all \( x > 2 \).

### Key Points:
- The graph starts at \( (-1, 0) \) and remains at \( y = 0 \) for \( x \leq -1 \).
- For \( -1 < x \leq 2 \), the function follows the line \( -x - 1 \), which passes through \( (-1, 0) \) and \( (2, -3) \).
- For \( x > 2 \), the function becomes \( y = -2 \), so it is a flat line.

Would you like me to create the graph for you? Also, here are some related questions:

1. What is the slope of the linear portion of \( f(x) \) for \( -1 < x \leq 2 \)?
2. How do we determine the continuity of piecewise functions at the points where the pieces change?
3. What are the limits of \( f(x) \) as \( x \to -1^+ \) and \( x \to 2^- \)?
4. How does the behavior of \( f(x) \) change as \( x \to \infty \)?
5. Can the function \( f(x) \) be expressed as a combination of Heaviside step functions?

**Tip:** When sketching piecewise functions, always check the values at the boundaries where the function definition changes.

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

Learn how to graph a piecewise function with both linear and constant segments. This step-by-step guide walks through sketching the graph for f(x), where f(x) = 0 for x <= -1, f(x) = -x - 1 for -1 < x <= 2, and f(x) = -2 for x > 2. Suitable for students in Grades 9-12.

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