The problem asks to find the piecewise function that represents the given graph. The graph consists of two line segments joined at $x = -2$. We need to find the equations of these line segments and the corresponding intervals for $x$.

1. Problem Description

The problem asks to find the piecewise function that represents the given graph. The graph consists of two line segments joined at $x = -2$. We need to find the equations of these line segments and the corresponding intervals for $x$.

2. Solution Steps

First, let's analyze the line segment for $x \ge -2$. Two points on this line are $(-2, -3)$ and $(2, 5)$.

The slope $m$ is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - (-3)}{2 - (-2)} = \frac{8}{4} = 2$

Using the point-slope form of a line, we have:

$y - y_1 = m(x - x_1)$

$y - (-3) = 2(x - (-2))$

$y + 3 = 2(x + 2)$

$y + 3 = 2x + 4$

$y = 2x + 4 - 3$

$y = 2x + 1$

So for $x \ge -2$, we have $f(x) = 2x + 1$.

Next, let's analyze the line segment for $x < -2$. Two points on this line are $(-7, 2)$ and $(-2, -3)$.

The slope $m$ is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 2}{-2 - (-7)} = \frac{-5}{-2 + 7} = \frac{-5}{5} = -1$

Using the point-slope form of a line, we have:

$y - y_1 = m(x - x_1)$

$y - (-3) = -1(x - (-2))$

$y + 3 = -1(x + 2)$

$y + 3 = -x - 2$

$y = -x - 2 - 3$

$y = -x - 5$

So for $x < -2$, we have $f(x) = -x - 5$.

Therefore, the piecewise function is:

$f(x) = \begin{cases} 2x + 1, & x \ge -2 \\ -x - 5, & x < -2 \end{cases}$

3. Final Answer

C. $f(x) = \begin{cases} 2x + 1 \text{ for } x \ge -2 \\ -x - 5 \text{ for } x < -2 \end{cases}$