Question 134359
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Let's break up this piecewise function into 3 pieces


Piece # 1: 

{{{y=4x+2}}}



Let's graph {{{y=4x+2}}} (tell me if you need help with graphing)


{{{graph(500,500,-10,10,-10,10,4x+2)}}}



Now let's <b>only</b> draw the part of the graph that is less than x=-2. In other words, erase everything to the right of x=-2 for the graph {{{y=4x+2}}}



{{{drawing(500,500,-10,10,-10,10,
graph(500,500,-10,10,-10,10,0),
circle(-10,-38,0.05),
line(-10,-38,-9.6,-36.4),
circle(-9.6,-36.4,0.05),
line(-9.6,-36.4,-9.2,-34.8),
circle(-9.2,-34.8,0.05),
line(-9.2,-34.8,-8.8,-33.2),
circle(-8.8,-33.2,0.05),
line(-8.8,-33.2,-8.4,-31.6),
circle(-8.4,-31.6,0.05),
line(-8.4,-31.6,-8,-30),
circle(-8,-30,0.05),
line(-8,-30,-7.6,-28.4),
circle(-7.6,-28.4,0.05),
line(-7.6,-28.4,-7.2,-26.8),
circle(-7.2,-26.8,0.05),
line(-7.2,-26.8,-6.8,-25.2),
circle(-6.8,-25.2,0.05),
line(-6.8,-25.2,-6.4,-23.6),
circle(-6.4,-23.6,0.05),
line(-6.4,-23.6,-6,-22),
circle(-6,-22,0.05),
line(-6,-22,-5.6,-20.4),
circle(-5.6,-20.4,0.05),
line(-5.6,-20.4,-5.2,-18.8),
circle(-5.2,-18.8,0.05),
line(-5.2,-18.8,-4.8,-17.2),
circle(-4.8,-17.2,0.05),
line(-4.8,-17.2,-4.4,-15.6),
circle(-4.4,-15.6,0.05),
line(-4.4,-15.6,-4,-14),
circle(-4,-14,0.05),
line(-4,-14,-3.6,-12.4),
circle(-3.6,-12.4,0.05),
line(-3.6,-12.4,-3.2,-10.8),
circle(-3.2,-10.8,0.05),
line(-3.2,-10.8,-2.8,-9.2),
circle(-2.8,-9.2,0.05),
line(-2.8,-9.2,-2.4,-7.6),
circle(-2.4,-7.6,0.05),
line(-2.4,-7.6,-2,-6),
circle(-2,-6,0.05)
)}}} Graph of {{{y=4x+2}}} with the condition that {{{x<-2}}}




<hr>


Piece # 2:



Now let's graph {{{y=x}}}




{{{graph(500,500,-10,10,-10,10,x)}}}



Now since we have the condition {{{-2<=x<=3}}}, this means that our new graph should look like:


{{{drawing(500,500,-10,10,-10,10,
graph(500,500,-10,10,-10,10,0),
circle(-1.6,-1.6,0.05),
line(-1.6,-1.6,-1.2,-1.2),
circle(-1.2,-1.2,0.05),
line(-1.2,-1.2,-0.799999999999996,-0.799999999999996),
circle(-0.799999999999996,-0.799999999999996,0.05),
line(-0.799999999999996,-0.799999999999996,-0.399999999999996,-0.399999999999996),
circle(-0.399999999999996,-0.399999999999996,0.05),
line(-0.399999999999996,-0.399999999999996,0,0),
circle(0,0,0.05),
line(0,0,0.400000000000004,0.400000000000004),
circle(0.400000000000004,0.400000000000004,0.05),
line(0.400000000000004,0.400000000000004,0.800000000000004,0.800000000000004),
circle(0.800000000000004,0.800000000000004,0.05),
line(0.800000000000004,0.800000000000004,1.2,1.2),
circle(1.2,1.2,0.05),
line(1.2,1.2,1.6,1.6),
circle(1.6,1.6,0.05),
line(1.6,1.6,2,2),
circle(2,2,0.05),
line(2,2,2.4,2.4),
circle(2.4,2.4,0.05),
line(2.4,2.4,2.8,2.8),
circle(2.8,2.8,0.05),
line(2.8,2.8,3.2,3.2)
)}}} Graph of {{{y=x}}} from x=-2 to x=3


<hr>



Piece #3 



Now let's graph {{{y=3x-1}}}




{{{graph(500,500,-10,10,-10,10,3x-1)}}}




Now because the corresponding inequality is {{{x>3}}}, this means that we <b>only</b> draw the part of the graph that is greater than 3



So the graph should look like



{{{drawing(500,500,-10,10,-10,10,
graph(500,500,-10,10,-10,10,0),
circle(3.2,8.6,0.05),
line(3.2,8.6,3.6,9.8),
circle(3.6,9.8,0.05),
line(3.6,9.8,4,11),
circle(4,11,0.05),
line(4,11,4.4,12.2),
circle(4.4,12.2,0.05),
line(4.4,12.2,4.80000000000001,13.4),
circle(4.80000000000001,13.4,0.05),
line(4.80000000000001,13.4,5.20000000000001,14.6),
circle(5.20000000000001,14.6,0.05),
line(5.20000000000001,14.6,5.60000000000001,15.8),
circle(5.60000000000001,15.8,0.05),
line(5.60000000000001,15.8,6.00000000000001,17),
circle(6.00000000000001,17,0.05),
line(6.00000000000001,17,6.40000000000001,18.2),
circle(6.40000000000001,18.2,0.05),
line(6.40000000000001,18.2,6.80000000000001,19.4),
circle(6.80000000000001,19.4,0.05),
line(6.80000000000001,19.4,7.20000000000001,20.6),
circle(7.20000000000001,20.6,0.05),
line(7.20000000000001,20.6,7.60000000000001,21.8),
circle(7.60000000000001,21.8,0.05),
line(7.60000000000001,21.8,8.00000000000001,23),
circle(8.00000000000001,23,0.05),
line(8.00000000000001,23,8.40000000000001,24.2),
circle(8.40000000000001,24.2,0.05),
line(8.40000000000001,24.2,8.80000000000001,25.4),
circle(8.80000000000001,25.4,0.05),
line(8.80000000000001,25.4,9.20000000000001,26.6),
circle(9.20000000000001,26.6,0.05),
line(9.20000000000001,26.6,9.60000000000001,27.8),
circle(9.60000000000001,27.8,0.05))}}} Graph of {{{y=3x-1}}} from x=3 to infinity




---------------------------------------



Now let's combine the three graphs and plot them on the same coordinate system



Graph of *[Tex \LARGE f(x)=\left\{ 4x+2 \text{ if x<-2} \\  x \text{ if -2\le x\le3} \\ 3x-1 \text{ if x>3}]



{{{drawing(500,500,-10,10,-10,10,
graph(500,500,-10,10,-10,10,0),
circle(-10,-38,0.05),
line(-10,-38,-9.6,-36.4),
circle(-9.6,-36.4,0.05),
line(-9.6,-36.4,-9.2,-34.8),
circle(-9.2,-34.8,0.05),
line(-9.2,-34.8,-8.8,-33.2),
circle(-8.8,-33.2,0.05),
line(-8.8,-33.2,-8.4,-31.6),
circle(-8.4,-31.6,0.05),
line(-8.4,-31.6,-8,-30),
circle(-8,-30,0.05),
line(-8,-30,-7.6,-28.4),
circle(-7.6,-28.4,0.05),
line(-7.6,-28.4,-7.2,-26.8),
circle(-7.2,-26.8,0.05),
line(-7.2,-26.8,-6.8,-25.2),
circle(-6.8,-25.2,0.05),
line(-6.8,-25.2,-6.4,-23.6),
circle(-6.4,-23.6,0.05),
line(-6.4,-23.6,-6,-22),
circle(-6,-22,0.05),
line(-6,-22,-5.6,-20.4),
circle(-5.6,-20.4,0.05),
line(-5.6,-20.4,-5.2,-18.8),
circle(-5.2,-18.8,0.05),
line(-5.2,-18.8,-4.8,-17.2),
circle(-4.8,-17.2,0.05),
line(-4.8,-17.2,-4.4,-15.6),
circle(-4.4,-15.6,0.05),
line(-4.4,-15.6,-4,-14),
circle(-4,-14,0.05),
line(-4,-14,-3.6,-12.4),
circle(-3.6,-12.4,0.05),
line(-3.6,-12.4,-3.2,-10.8),
circle(-3.2,-10.8,0.05),
line(-3.2,-10.8,-2.8,-9.2),
circle(-2.8,-9.2,0.05),
line(-2.8,-9.2,-2.4,-7.6),
circle(-2.4,-7.6,0.05),
line(-2.4,-7.6,-2,-6),
circle(-2,-6,0.05),
circle(-1.6,-1.6,0.05),
line(-1.6,-1.6,-1.2,-1.2),
circle(-1.2,-1.2,0.05),
line(-1.2,-1.2,-0.799999999999996,-0.799999999999996),
circle(-0.799999999999996,-0.799999999999996,0.05),
line(-0.799999999999996,-0.799999999999996,-0.399999999999996,-0.399999999999996),
circle(-0.399999999999996,-0.399999999999996,0.05),
line(-0.399999999999996,-0.399999999999996,0,0),
circle(0,0,0.05),
line(0,0,0.400000000000004,0.400000000000004),
circle(0.400000000000004,0.400000000000004,0.05),
line(0.400000000000004,0.400000000000004,0.800000000000004,0.800000000000004),
circle(0.800000000000004,0.800000000000004,0.05),
line(0.800000000000004,0.800000000000004,1.2,1.2),
circle(1.2,1.2,0.05),
line(1.2,1.2,1.6,1.6),
circle(1.6,1.6,0.05),
line(1.6,1.6,2,2),
circle(2,2,0.05),
line(2,2,2.4,2.4),
circle(2.4,2.4,0.05),
line(2.4,2.4,2.8,2.8),
circle(2.8,2.8,0.05),
line(2.8,2.8,3.2,3.2),
circle(3.2,8.6,0.05),
line(3.2,8.6,3.6,9.8),
circle(3.6,9.8,0.05),
line(3.6,9.8,4,11),
circle(4,11,0.05),
line(4,11,4.4,12.2),
circle(4.4,12.2,0.05),
line(4.4,12.2,4.80000000000001,13.4),
circle(4.80000000000001,13.4,0.05),
line(4.80000000000001,13.4,5.20000000000001,14.6),
circle(5.20000000000001,14.6,0.05),
line(5.20000000000001,14.6,5.60000000000001,15.8),
circle(5.60000000000001,15.8,0.05),
line(5.60000000000001,15.8,6.00000000000001,17),
circle(6.00000000000001,17,0.05),
line(6.00000000000001,17,6.40000000000001,18.2),
circle(6.40000000000001,18.2,0.05),
line(6.40000000000001,18.2,6.80000000000001,19.4),
circle(6.80000000000001,19.4,0.05),
line(6.80000000000001,19.4,7.20000000000001,20.6),
circle(7.20000000000001,20.6,0.05),
line(7.20000000000001,20.6,7.60000000000001,21.8),
circle(7.60000000000001,21.8,0.05),
line(7.60000000000001,21.8,8.00000000000001,23),
circle(8.00000000000001,23,0.05),
line(8.00000000000001,23,8.40000000000001,24.2),
circle(8.40000000000001,24.2,0.05),
line(8.40000000000001,24.2,8.80000000000001,25.4),
circle(8.80000000000001,25.4,0.05),
line(8.80000000000001,25.4,9.20000000000001,26.6),
circle(9.20000000000001,26.6,0.05),
line(9.20000000000001,26.6,9.60000000000001,27.8),
circle(9.60000000000001,27.8,0.05)



)}}}