To solve the problem of writing a piecewise-defined function for the given graph, we need to carefully observe the two distinct line segments and their respective slopes.

Steps:

1. Identify the segments:

   • The first line segment starts from point $(-4, 1)$ and goes to $(-1, -2)$.
   • The second line segment starts from point $(-1, -2)$ and goes to $(2, 1)$.

2. Find the equations of each line:

   • First segment $(-4, 1)$ to $(-1, -2)$:

     • The slope (m) is given by: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 1}{-1 - (-4)} = \frac{-3}{3} = -1$
     • Using the point-slope form $y - y_1 = m(x - x_1)$ with the point $(-1, -2)$: $y + 2 = -1(x + 1) \implies y = -x - 1$
     • Therefore, the equation of the first segment is $y = -x - 1$.

   • Second segment $(-1, -2)$ to $(2, 1)$:

     • The slope (m) is given by: $m = \frac{1 - (-2)}{2 - (-1)} = \frac{3}{3} = 1$
     • Using the point-slope form with point $(-1, -2)$: $y + 2 = 1(x + 1) \implies y = x - 1$
     • Therefore, the equation of the second segment is $y = x - 1$.

3. Write the piecewise function: The function can be written as:

   x - 1 & \text{for } x \geq -1 \\ -x - 1 & \text{for } x < -1 \end{cases}$$

Conclusion:

The correct answer is option D: