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The problem you provided involves graphing a piecewise-defined function, which is given as follows:

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

### To graph this piecewise function:

1. **For \(x \leq -2\)**: The function is constant and equal to 1. So for all values of \(x \leq -2\), the graph will be a horizontal line at \(y = 1\).
   
2. **For \(-2 < x \leq 2\)**: The function is equal to \(x\), which means the graph will be the line \(y = x\) for this interval. This is a diagonal line passing through the origin, but restricted between \(x = -2\) and \(x = 2\).

3. **For \(x > 2\)**: The function is again constant and equal to 1, so for all values of \(x > 2\), the graph will be a horizontal line at \(y = 1\).

### Key points to plot:
- **At \(x = -2\)**: The value is \(g(-2) = 1\). So, there is a closed circle at \((-2, 1)\).
- **At \(x = 2\)**: The value is \(g(2) = 2\). So, there is a closed circle at \((2, 2)\).
- **For \(x > 2\)** and \(x \leq -2\)**: The graph is a flat line at \(y = 1\) on both sides.

### Graphing Strategy:
- Plot a horizontal line at \(y = 1\) for \(x \leq -2\) and \(x > 2\).
- Plot a diagonal line with slope 1 between \(x = -2\) and \(x = 2\).
- Make sure to use solid circles at the transition points for continuous functions, such as at \(x = -2\) and \(x = 2\).

Would you like a detailed plot or any further explanation?

Here are 5 follow-up questions:
1. How do you determine the transition points in piecewise functions?
2. Why is there a closed circle at \(x = 2\) and \(x = -2\)?
3. Can you describe the behavior of the graph as \(x\) approaches -2 and 2?
4. What changes would occur if the intervals were strict inequalities (e.g., \(x < -2\))?
5. How does the function behave for large values of \(x\)?

**Tip:** In piecewise functions, always check for continuity at the transition points to ensure there are no gaps in the graph.

Suppose that the function g is defined, for all real numbers, as follows:

g(x) = 
{
1 if x ≤ -2
x if -2 < x ≤ 2
1 if x > 2
}

Graph the function g.

Learn how to graph a piecewise-defined function involving three intervals: constant values for x ≤ -2 and x > 2, and a linear function for -2 < x ≤ 2. Understand key concepts such as transition points and continuity for high school math students.

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