Let's analyze the given graph to write the piecewise function.

Steps to solve:

1. The graph is made of two line segments with different slopes.

2. Identify the equations of the two lines:

   • For $x \geq -2$: This segment runs from point $(-2, -5)$ to $(5, 2)$. We need to find the slope and the equation.
   • For $x < -2$: This segment runs from $(-4, 4)$ to $(-2, -5)$. Similarly, we find the slope and equation.

Step 1: Equation for $x \geq -2$

The two points on this line are $(-2, -5)$ and $(5, 2)$.

• Slope: $m = \frac{2 - (-5)}{5 - (-2)} = \frac{7}{7} = 1$

• Equation: Using point-slope form $y - y_1 = m(x - x_1)$, with point $(-2, -5)$: $y + 5 = 1(x + 2)$ Simplifying: $y = x - 3$

Thus, for $x \geq -2$, the equation is $f(x) = x - 3$.

Step 2: Equation for $x < -2$

The two points on this line are $(-4, 4)$ and $(-2, -5)$.

• Slope: $m = \frac{-5 - 4}{-2 - (-4)} = \frac{-9}{2}$

• Equation: Using point-slope form with point $(-2, -5)$: $y + 5 = -\frac{9}{2}(x + 2)$ Simplifying: $y = -\frac{9}{2}(x + 2) - 5$ Expand: $y = -\frac{9}{2}x - 9 - 5$ $y = -\frac{9}{2}x - 14$

Thus, for $x < -2$, the equation is $f(x) = -\frac{9}{2}x - 14$.

Final Piecewise Function: