The image you uploaded shows a mathematical problem involving the calculation of a piecewise function based on the graph provided.

Step-by-Step Solution:

1. Understanding the Graph:

   • The graph displays two line segments. Each segment represents a portion of a piecewise function.
   • The first segment starts at the point $(-4, 0)$ and ends at $(2, 6)$.
   • The second segment starts at $(2, 6)$ and ends at $(6, 2)$.

2. Defining the Piecewise Function:

   • For $x$ values from $-4$ to $2$ (inclusive), the function is defined by the first line segment.
   • For $x$ values from $2$ to $6$ (inclusive), the function is defined by the second line segment.

3. Finding the Equation of Each Line Segment:

   • First Segment (from $(-4, 0)$ to $(2, 6)$):

     • The slope $m_1$ of the line can be calculated as: $m_1 = \frac{6 - 0}{2 - (-4)} = \frac{6}{6} = 1$
     • Using the point-slope form $y - y_1 = m(x - x_1)$, with point $(-4, 0)$: $y = 1(x + 4) = x + 4$
     • So, for $x$ from $-4$ to $2$: $f(x) = x + 4$

   • Second Segment (from $(2, 6)$ to $(6, 2)$):

     • The slope $m_2$ of the line is: $m_2 = \frac{2 - 6}{6 - 2} = \frac{-4}{4} = -1$
     • Using the point-slope form $y - y_1 = m(x - x_1)$, with point $(2, 6)$: $y - 6 = -1(x - 2)$ $y = -x + 2 + 6 = -x + 8$
     • So, for $x$ from $2$ to $6$: $f(x) = -x + 8$

Final Piecewise Function:

The piecewise function based on the given graph is: