https://file.aiquickdraw.com/mathbot/file/1725505504150bo59w6ru.jpg

The image shows a graph that is piecewise linear, with two distinct line segments. Let’s analyze the graph step by step to express the function as a piecewise function:

### Analyzing the graph:
1. **First segment**:
   - This line starts at \((-2, -1)\) and goes horizontally to \((0, -1)\).
   - It is a horizontal line with \( y = -1 \) for \( x \in [-2, 0] \).

2. **Second segment**:
   - The second line segment starts at \((0, 0)\) and passes through \((2, 2)\).
   - It has a positive slope and is a linear function.
   - Using the slope formula \(\frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 0}{2 - 0} = 1\), we conclude the slope is \(1\).
   - The equation of the line is \( y = x \) for \( x \in (0, 2] \).

### Piecewise Function Representation:
Thus, the piecewise function for this graph is:
\[
f(x) =
\begin{cases}
-1 & \text{if } -2 \leq x \leq 0, \\
x & \text{if } 0 < x \leq 2.
\end{cases}
\]

Would you like further details on how to derive the equation of the second segment, or do you have any questions about the representation?

Here are 5 related questions to expand your understanding:
1. How do you calculate the slope of a line segment given two points?
2. What are the conditions for continuity in piecewise functions?
3. Can a piecewise function have more than two segments, and how would that affect the function's behavior?
4. How can you determine the domain and range of a piecewise function?
5. What would happen if the points where the pieces meet were not continuous?

**Tip:** When dealing with piecewise functions, always check whether the function is continuous at the transition points between different segments. This helps in analyzing the overall behavior of the function.

Express the function graphed as a piecewise function.

Explore how to express a graphed piecewise linear function in its mathematical form. Learn to identify linear segments, calculate slopes, and represent the function using piecewise notation. This example is suitable for students in Grades 8-10.

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