The image presents a graph and a piecewise function problem. The graph is composed of two linear segments: one with a negative slope from $(-4,1)$ to $(0,-3)$, and one with a positive slope from $(0,-3)$ to $(4,5)$. The task is to write the piecewise function that represents the graph and select the correct option from four given choices.

Step-by-step analysis:

First line segment ($-4, 1$ to $(0, -3)$):

• This segment has a negative slope.
• To find the equation of the line, calculate the slope $m$: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 1}{0 - (-4)} = \frac{-4}{4} = -1$
• The equation of the line is in the form $y = mx + b$, where $m = -1$ and $b$ is the y-intercept.
• Using the point $(0, -3)$, substitute to find $b$: $-3 = -1(0) + b \implies b = -3$
• The equation of this line is $y = -x - 3$, valid for $x < 0$.

Second line segment ($0, -3$ to $4, 5$):

• This segment has a positive slope.
• Calculate the slope $m$: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - (-3)}{4 - 0} = \frac{8}{4} = 2$
• The equation of the line is $y = 2x + b$. Using the point $(0, -3)$: $-3 = 2(0) + b \implies b = -3$
• The equation of this line is $y = 2x - 3$, valid for $x \geq 0$.

Final piecewise function: