https://file.aiquickdraw.com/mathbot/file/1726195068004czomf2ol.jpg

The given graph represents a piecewise-defined function, and part of the function is already provided. Let's analyze each piece of the function:

1. **For \( x \leq -1 \)**: The graph is a horizontal line at \( y = 4 \), so the function is constant:
   \[
   f(x) = 4
   \]
   This is already correctly provided.

2. **For \( -1 < x \leq 2 \)**: The function is linear, and the expression provided is \( f(x) = 4 - x \), which is correct since this line has a slope of -1 (it decreases by 1 unit for each increase in \( x \)) and the y-intercept for this line appears to be 4.

3. **For \( x > 2 \)**: The function becomes a horizontal line at \( y = 1 \). This means that for all values of \( x > 2 \), the function takes the value of 1.

Thus, the missing part of the piecewise function for \( x > 2 \) is:
\[
f(x) = 1
\]

### Final piecewise function:
\[
f(x) =
\begin{cases} 
4 & \text{if } x \leq -1 \\
4 - x & \text{if } -1 < x \leq 2 \\
1 & \text{if } x > 2
\end{cases}
\]

Would you like further details on how this function was derived or any other clarifications?

Here are some follow-up questions you might consider:
1. How do we derive equations from the slopes of linear segments?
2. Why is the value constant beyond \( x = 2 \)?
3. What does the slope tell us about the behavior of the function?
4. Could there be any alternative ways to represent this function?
5. How would the graph look if the values were different for each piece?

**Tip:** When analyzing a graph, always check for horizontal lines, as they correspond to constant function values over that interval.

A graph of a piecewise-defined function is given, and the user is asked to find a formula for the function in three intervals: x ≤ -1, -1 < x ≤ 2, and x > 2.

Learn how to define a piecewise function from a graph with three intervals. The function includes a constant value for x ≤ -1, a linear equation for -1 < x ≤ 2, and another constant for x > 2. Suitable for high school students.

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