https://file.aiquickdraw.com/mathbot/file/1726194951024er3xp6x1.jpg

The problem asks to write the piecewise function whose graph is shown in the image.

### Analysis of the Graph:

1. **Leftmost segment** (for \( x \leq -2 \)): 
   The line is sloped downwards and passes through the point \((-2, 0)\) and continues in the negative direction. This line has a slope of -1, indicating the equation should be \( f(x) = -x - 2 \) for this interval.

2. **Middle segment** (for \( -2 < x \leq 0 \)):
   The line passes through the point \((0, 2)\) and continues until \((0, 2)\). This part of the graph looks like a straight line with slope 1. The equation for this segment is \( f(x) = x + 2 \).

3. **Rightmost segment** (for \( 0 < x \leq 2 \)): 
   This is a quadratic-looking curve, and based on the shape and points it passes through, we can guess it's part of a downward parabola. But as it is not completely clear, we rely on the stated function \( f(x) = x - 2 \) for this interval.

### The piecewise function is as follows:
\[
f(x) =
\begin{cases} 
x - 2, & x = 2 \\
x + 2, & -2 \leq x \leq 0 \\
x - 2, & 0 \leq x \leq 2
\end{cases}
\]

Let me know if you'd like more details or further explanation!

Here are some follow-up questions related to piecewise functions:

1. What conditions must be true for a function to be continuous at a point?
2. How do you determine the equation of a line from two given points?
3. What is the general form of a quadratic equation, and how can you recognize it from a graph?
4. Can piecewise functions have discontinuities, and what types of discontinuities exist?
5. How do the domain and range of a piecewise function differ from a regular function?

**Tip**: When sketching graphs, pay close attention to the slope and y-intercepts of the lines in each piece of the function.

Write the piecewise function whose graph is shown in the image.

Learn how to write a piecewise function from a graph. This problem focuses on interpreting linear and non-linear segments in a piecewise graph, suitable for students in Grades 9-11.

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