https://file.aiquickdraw.com/mathbot/file/1727412610964necp1eir.jpg

The graph shows three distinct segments that need to be expressed as a piecewise function:

1. **For \(x < -2\)**:  
   The graph is a horizontal line with a constant value at \( y = -2 \).

2. **For \( -2 \leq x \leq 0 \)**:  
   The graph is another horizontal line at \( y = 0 \). However, there is an open circle at \(x = 0\), indicating that the value is not included at that point. The notation should reflect this discontinuity at \(x = 0\).

3. **For \(x > 0\)**:  
   The graph is a line with positive slope, passing through the point \( (0, 1) \) and extending upwards. To find the equation of this line, we can use the points \( (0, 1) \) and \( (2, 2) \) to determine the slope:
   \[
   \text{slope} = \frac{2 - 1}{2 - 0} = \frac{1}{2}.
   \]
   Using the point-slope form of a line, the equation is:
   \[
   y - 1 = \frac{1}{2}(x - 0) \quad \Rightarrow \quad y = \frac{1}{2}x + 1.
   \]

Now, let's write the piecewise function:

\[
f(x) = \begin{cases} 
-2, & x < -2, \\
0, & -2 \leq x < 0, \\
\frac{1}{2}x + 1, & x > 0.
\end{cases}
\]

This is the correct piecewise function for the given graph.

Would you like more details or have any questions?

### Related Questions:
1. How do you handle open and closed circles when writing piecewise functions?
2. How do you find the equation of a line given two points on a graph?
3. What is the difference between continuous and discontinuous functions in terms of graph behavior?
4. Can a piecewise function have overlapping intervals for different pieces?
5. How do you determine the domain of a piecewise function from its graph?

### Tip:
Always check whether each segment of the piecewise function includes or excludes endpoints using open or closed circles.

Write down the piecewise function for the graph given below. Make sure to use correct format and notations.

This problem explores how to write a piecewise function from a graph that consists of both constant and linear segments. Learn to use correct notations, handle open and closed circles, and derive the equation of a line from two points. Suitable for Grades 9-11 students.

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