Answer:
The piecewise function is given below as
[tex]f(x)=\begin{cases}(x+3)^2+2;x\leq-2 \\ -2x+4,;-2
Step 1:The first function is given below as
[tex]\begin{gathered} f(x)=(x+3)^2+2 \\ \text{when x=2} \\ f(2)=(-2+3)^2+2 \\ f(2)=1^2+2 \\ f(2)=1+2=3 \\ (-2,3) \end{gathered}[/tex]
Hence, the graph above is a parabola that is opened upwards because the coefficient of the leading term is greater than 1, and it will be shaded at point (-2,3) because of the less than or equal to sign
(it has a negative slope that is as x-increase, y decreases, and vice versa)
Step 2:
The second function is given below as
[tex]\begin{gathered} f(x)=-2x+4 \\ \text{when x=-2} \\ f(-2)=-2(-2)+4 \\ f(-2)=4+4 \\ f(-2)=8 \\ (-2,8) \\ \\ f(x)=-2x+4 \\ \text{when x=-4} \\ f(-4)=-2(-4)+4 \\ f(-4)=8+4 \\ f(-4)=12 \\ (-4,12) \end{gathered}[/tex]
The second equation is a straight line that will be open at points (-2,8) and (4,12) because of the strictly less than sign
Step 3:
The thrid function is
[tex]\begin{gathered} f(x)=3x-15;x\ge4 \\ \text{when x=4} \\ f(4)=3(4)-15 \\ f(4)=12-15 \\ f(4)=-3 \\ (-3,4) \end{gathered}[/tex]
The equation above is a straight line and will be shaded at point (-3,4) because of the greater than or equal to sign
(it has a positive slope that is as x increases,y also increases and vice versa)
Hence,
The final answer will be
