https://file.aiquickdraw.com/mathbot/file/1728063425589afrgxe52.jpg

The problem asks to identify the graph that corresponds to the given piecewise-defined function:

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

### Breaking down the function:
1. **For \( x \leq -2 \):** The function is \( f(x) = -\frac{1}{2}x - 1 \). This is a line with a slope of \(-\frac{1}{2}\) and a y-intercept of \(-1\). It applies for all \( x \) values less than or equal to \(-2\).
   
2. **For \( -2 < x < 1 \):** The function is constant, \( f(x) = 1 \), meaning the graph will be a horizontal line at \( y = 1 \) between \( x = -2 \) and \( x = 1 \) (excluding the boundaries).

3. **For \( x \geq 1 \):** The function is \( f(x) = -x \), a line with a slope of \(-1\), starting at \( x = 1 \) and extending rightward.

### Analyzing the options:
We need to check which graph correctly matches this behavior:

- The graph should:
  1. Have a line with slope \(-\frac{1}{2}\) for \( x \leq -2 \).
  2. Show a horizontal line at \( y = 1 \) between \( -2 < x < 1 \).
  3. Display a line with slope \(-1\) for \( x \geq 1 \).

After analyzing, **Option C** shows the correct combination of slopes and constant values based on the piecewise function definition.

### Final answer: **C**

Do you need any further clarifications or details? 

Here are some related questions you can explore:
1. How do you determine the slope and intercept from the equation of a line?
2. What happens if we change the intervals in a piecewise function?
3. How do you graph a function with multiple intervals?
4. Why are open and closed circles used in piecewise function graphs?
5. How can you determine continuity from a piecewise function graph?

**Tip:** When graphing piecewise functions, always pay attention to the open and closed circles to correctly represent the boundaries of the function in each interval.

Graph the piecewise function provided in the image and choose the correct graph.

Learn how to graph the piecewise function f(x) with different expressions for three intervals. Analyze the graph for f(x) = -1/2 * x - 1 (x <= -2), f(x) = 1 (-2 < x < 1), and f(x) = -x (x >= 1). Suitable for high school students.

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