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To graph the function \(f(x)\) defined by the following piecewise function:

\[
f(x) = 
\begin{cases} 
3 & \text{if } x < 0 \\
-3 & \text{if } x = 0 \\
2 & \text{if } x > 0 
\end{cases}
\]

We need to follow these steps:

1. **Draw the horizontal lines for each condition**:
   - For \(x < 0\): Draw a horizontal line at \(y = 3\). This line should continue infinitely to the left but should stop just before reaching \(x = 0\).
   - For \(x > 0\): Draw a horizontal line at \(y = 2\). This line should continue infinitely to the right but should stop just after \(x = 0\).

2. **Plot the point for \(x = 0\)**:
   - At \(x = 0\), the function value is \(-3\). So, plot a point at \((0, -3)\). This point should be a filled-in circle, indicating that the function equals \(-3\) exactly at \(x = 0\).

### Graph Construction:

- **Left of the origin**: Draw a solid line parallel to the x-axis at \(y = 3\) extending leftward from \(x = 0\) without including \(x = 0\). Use an open circle (unfilled dot) at \((0, 3)\) to show that the function does not include this value at \(x = 0\).
- **At the origin**: Place a filled dot at \((0, -3)\).
- **Right of the origin**: Draw a solid line parallel to the x-axis at \(y = 2\) extending rightward from \(x = 0\) without including \(x = 0\). Use an open circle (unfilled dot) at \((0, 2)\) to indicate that this part does not include the point at \(x = 0\).

### Final Notes:
- Ensure the horizontal lines are distinctly separated at \(y = 3\) for \(x < 0\) and \(y = 2\) for \(x > 0\).
- Make sure the point at \(x = 0\) is clearly marked at \(-3\).

Would you like to see a sketch of this graph, or do you have any further questions?

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Here are 5 related questions you might find useful:

1. What is a piecewise function, and how is it defined?
2. How do you determine the continuity of a piecewise function?
3. What is the difference between open and closed circles when graphing piecewise functions?
4. How can you determine the domain and range of a piecewise function?
5. How would the graph change if one of the piecewise conditions had a different inequality, like \(x \leq 0\)?

**Tip:** When graphing piecewise functions, always carefully observe the conditions and include or exclude endpoints as specified.

Learn how to graph the piecewise function f(x) = 3 for x < 0, f(x) = -3 for x = 0, f(x) = 2 for x > 0 step-by-step. Includes detailed instructions on drawing horizontal lines and plotting points. Suitable for high school students.

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