https://file.aiquickdraw.com/mathbot/file/1728064464477o6p0u91z.jpg

Let's analyze the piecewise-defined function based on the given graph.

The graph has three important points:
- (-5, 2)
- (0, -3)
- (3, 3)

It has two segments, one for \(x < 0\) and one for \(x \geq 0\).

### 1. **For \(x < 0\) (left side)**:
The line passes through the points (-5, 2) and (0, -3). To find the equation of the line, we first determine the slope:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 2}{0 - (-5)} = \frac{-5}{5} = -1
\]

Now, using the point-slope form of a line \(y - y_1 = m(x - x_1)\), with slope \(m = -1\) and point \((0, -3)\), the equation becomes:

\[
y = -1(x - 0) - 3 \quad \text{or} \quad y = -x - 3
\]

Thus, for \(x < 0\), the equation is:
\[
f(x) = -x - 3
\]

### 2. **For \(x \geq 0\) (right side)**:
The line passes through the points (0, -3) and (3, 3). The slope is:

\[
\text{slope} = \frac{3 - (-3)}{3 - 0} = \frac{6}{3} = 2
\]

Using the point-slope form with the point (0, -3) and slope 2:

\[
y - (-3) = 2(x - 0) \quad \text{or} \quad y = 2x - 3
\]

Thus, for \(x \geq 0\), the equation is:
\[
f(x) = 2x - 3
\]

### Piecewise Function:
\[
f(x) = 
\begin{cases} 
- x - 3 & \text{for} \ x < 0 \\
2x - 3 & \text{for} \ x \geq 0
\end{cases}
\]

### Correct Answer:
The correct option is **C**.

Would you like a more detailed explanation on any part of this?

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Here are some related questions:
1. How do you find the slope between two points?
2. What is the point-slope form of a line equation?
3. What is a piecewise function and when is it used?
4. How do you handle different cases in piecewise functions?
5. How can you interpret graphs to find equations?

**Tip:** When working with piecewise functions, always check the boundary values carefully to ensure you apply the correct equation for each interval.

Write a piecewise function for the graph shown, and choose the correct answer from the options provided.

Learn how to write a piecewise-defined function based on a given graph. This solution involves calculating the slope of a line and using point-slope form to create the function, with the final result represented as a piecewise function. Suitable for students in Grades 9-12.

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