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The graph you've provided shows a piecewise function, which is split into two parts based on different intervals of \(x\). Here's how we can determine each part of the piecewise function:

1. **Left part of the graph (for \(x \leq -2\)):**
   - The line goes from the point \((-9, 4)\) to \((-2, -2)\).
   - We can find the slope \(m\) of the line using the slope formula:
     \[
     m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 4}{-2 - (-9)} = \frac{-6}{7}
     \]
   - So, the slope is \(m = -\frac{6}{7}\).
   - Using point-slope form with the point \((-2, -2)\):
     \[
     y - (-2) = -\frac{6}{7}(x - (-2))
     \]
     Simplifying:
     \[
     y + 2 = -\frac{6}{7}(x + 2)
     \]
     \[
     y = -\frac{6}{7}(x + 2) - 2
     \]
     So, the equation of the line for this part is:
     \[
     y = -\frac{6}{7}x - \frac{10}{7}
     \]
   - This equation holds for \(x \leq -2\).

2. **Right part of the graph (for \(x > -2\)):**
   - The line goes from the point \((-2, -2)\) to \((3, 3)\).
   - Again, we can find the slope \(m\):
     \[
     m = \frac{3 - (-2)}{3 - (-2)} = \frac{5}{5} = 1
     \]
   - Using point-slope form with the point \((-2, -2)\):
     \[
     y - (-2) = 1(x - (-2))
     \]
     Simplifying:
     \[
     y + 2 = x + 2
     \]
     \[
     y = x
     \]
   - This equation holds for \(x > -2\).

### Final Piecewise Function:

\[
f(x) = 
\begin{cases} 
-\frac{6}{7}x - \frac{10}{7} & \text{for } x \leq -2 \\
x & \text{for } x > -2
\end{cases}
\]

Let me know if you need more details or have questions!

### Related Questions:
1. How do you find the slope of a line between two points?
2. What is the point-slope form of a line?
3. How can we interpret the open and closed dots on the graph?
4. What is the general process for writing a piecewise function?
5. How do you solve a system of equations for the intersection points of two lines?

**Tip:** When determining piecewise functions from a graph, always observe the boundaries carefully to ensure the correct form of inequalities.

Express the function graphed on the axes below as a piecewise function.

Learn how to express a graphed function as a piecewise function. This problem involves calculating the slope between points, applying linear equations, and writing the final piecewise function for different intervals of x.

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