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The graph shows a piecewise-defined function. We need to determine the correct function for \( f(x) \) based on the two segments shown:

1. **First segment**: The line starts at point \( (-1, -2) \) and goes up to \( (2, 3) \). To determine its equation, calculate the slope:
   \[
   \text{slope} = \frac{3 - (-2)}{2 - (-1)} = \frac{5}{3}.
   \]
   Using point-slope form \( y - y_1 = m(x - x_1) \) with the point \( (-1, -2) \) and slope \( \frac{5}{3} \), we get:
   \[
   y + 2 = \frac{5}{3}(x + 1).
   \]
   Simplifying:
   \[
   y = \frac{5}{3}x + \frac{5}{3} - 2 = \frac{5}{3}x - \frac{1}{3}.
   \]

   Therefore, for \( -1 \leq x \leq 2 \), the equation is \( f(x) = \frac{5}{3}x - \frac{1}{3} \).

2. **Second segment**: The line starts at \( (2, 3) \) and extends horizontally to the right. Since it's a horizontal line, its equation is simply \( y = 3 \) for \( x \geq 2 \).

Thus, the piecewise function is:
\[
f(x) =
\begin{cases}
\frac{5}{3}x - \frac{1}{3}, & \text{if } -1 \leq x \leq 2, \\
3, & \text{if } x > 2.
\end{cases}
\]

This matches option **B** in the answer choices.

Would you like further details or explanations? Here are five related questions to consider:

1. How is the slope of a line calculated?
2. What is the point-slope form of a linear equation?
3. How do you find the equation of a line passing through two points?
4. What is the equation of a horizontal line?
5. How does the domain of each piece affect the form of a piecewise function?

**Tip**: When finding a line’s equation from a graph, always start by calculating the slope using two points!

Give the rule that describes the piecewise-defined function in the graph below.

Learn how to write a piecewise-defined function by analyzing a graph. This problem involves finding the slope, using the point-slope form, and identifying equations of linear and horizontal lines. Suitable for high school students in Grades 9-12.

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