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

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

### Steps to Sketch the Graph:

1. **For \( x \leq -2 \):**
   - The function is constant, \( f(x) = -3 \). 
   - Plot a horizontal line at \( y = -3 \) for all \( x \leq -2 \).
   - Since \( f(x) = -3 \) is valid for \( x = -2 \) as well, use a **closed dot** at \( (-2, -3) \).

2. **For \( -2 < x \leq 2 \):**
   - The function follows \( f(x) = x \), which is a linear function with a slope of 1.
   - Plot this line for the range \( -2 < x \leq 2 \). 
   - At \( x = -2 \), the function does not include the point, so use an **open dot** at \( (-2, -2) \).
   - At \( x = 2 \), the function includes the point, so use a **closed dot** at \( (2, 2) \).

3. **For \( x > 2 \):**
   - The function is constant, \( f(x) = 4 \).
   - Plot a horizontal line at \( y = 4 \) for all \( x > 2 \).
   - Use an **open dot** at \( (2, 4) \), since \( f(x) = 4 \) does not include \( x = 2 \).

### Key Points to Include:
- Horizontal line at \( y = -3 \) for \( x \leq -2 \), with a closed dot at \( (-2, -3) \).
- Line \( y = x \) from \( x = -2 \) to \( x = 2 \), with an open dot at \( (-2, -2) \) and a closed dot at \( (2, 2) \).
- Horizontal line at \( y = 4 \) for \( x > 2 \), with an open dot at \( (2, 4) \).

This should give you a clear graph of the piecewise function.

Let me know if you want further details or clarifications! Here are some related questions:

1. What is the domain of the function?
2. How do you calculate the range of a piecewise function?
3. Can piecewise functions be continuous?
4. What is the significance of open and closed dots in graphing?
5. How can we identify if a piecewise function is differentiable?

**Tip:** Always check for discontinuities at the break points of piecewise functions!

Sketch a graph of the piecewise function f(x) = { -3 if x <= -2, x if -2 < x <= 2, 4 if x > 2 }.

Learn how to graph the piecewise function f(x), defined as -3 for x <= -2, x for -2 < x <= 2, and 4 for x > 2. This problem involves graphing constant and linear functions with open and closed points. Suitable for Grades 8-10.

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