1. The problem is to find the piecewise linear function $F(x)$ that passes through the points $(-4,0)$, $(-2,0)$, $(0,3)$, $(2,2)$, and $(4,0)$. 2. Since the function is piecewise linear, it consists of line segments connecting these points. We will find the equation of each line segment using the slope formula and point-slope form. 3. For the interval $[-4,-2]$, the points are $(-4,0)$ and $(-2,0)$. The slope is $$m=\frac{0-0}{-2-(-4)}=\frac{0}{2}=0.$$ The line is horizontal: $$F(x)=0 \text{ for } -4 \leq x \leq -2.$$ 4. For the interval $[-2,0]$, the points are $(-2,0)$ and $(0,3)$. The slope is $$m=\frac{3-0}{0-(-2)}=\frac{3}{2}.$$ Using point-slope form with point $(-2,0)$: $$y-0=\frac{3}{2}(x+2) \Rightarrow F(x)=\frac{3}{2}x+3 \text{ for } -2 < x \leq 0.$$ 5. For the interval $[0,2]$, the points are $(0,3)$ and $(2,2)$. The slope is $$m=\frac{2-3}{2-0}=\frac{-1}{2}=-\frac{1}{2}.$$ Using point-slope form with point $(0,3)$: $$y-3=-\frac{1}{2}(x-0) \Rightarrow F(x)=-\frac{1}{2}x+3 \text{ for } 0 < x \leq 2.$$ 6. For the interval $[2,4]$, the points are $(2,2)$ and $(4,0)$. The slope is $$m=\frac{0-2}{4-2}=\frac{-2}{2}=-1.$$ Using point-slope form with point $(2,2)$: $$y-2=-1(x-2) \Rightarrow F(x)=-x+4 \text{ for } 2 < x \leq 4.$$ 7. The piecewise function is therefore: $$F(x)=\begin{cases} 0 & -4 \leq x \leq -2 \\ \frac{3}{2}x+3 & -2 < x \leq 0 \\ -\frac{1}{2}x+3 & 0 < x \leq 2 \\ -x+4 & 2 < x \leq 4 \end{cases}$$