1. **State the problem:** We need to write the piecewise function $f(x)$ based on the given graph description. 2. **Analyze the first part:** The first part is a line segment from $(-8,-2)$ to $(-2,4)$, both points included (solid points). 3. **Find the slope of the line segment:** $$m = \frac{4 - (-2)}{-2 - (-8)} = \frac{6}{6} = 1$$ 4. **Find the equation of the line:** Using point-slope form with point $(-8,-2)$: $$y - (-2) = 1(x - (-8))$$ $$y + 2 = x + 8$$ $$y = x + 6$$ 5. **Domain for the first part:** Since the segment is from $x = -8$ to $x = -2$ inclusive, the domain is $-8 \leq x \leq -2$. 6. **Analyze the second part:** The second part is a horizontal ray starting at $(-2,-4)$ but the point is empty (not included), extending to the right indefinitely. 7. **Equation for the second part:** Since it is horizontal at $y = -4$, the function is: $$f(x) = -4$$ 8. **Domain for the second part:** Since it starts just after $x = -2$ (not included) and extends to infinity, the domain is $x > -2$. 9. **Write the piecewise function:** $$ f(x) = \begin{cases} x + 6 & \text{for } -8 \leq x \leq -2 \\ -4 & \text{for } x > -2 \end{cases} $$