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: