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- WELCOME TO THE FIRST EXAMPLE OF GRAPHING
A PIECEWISE DEFINED FUNCTION.
LOOKING AT OUR PIECE DEFINED FUNCTION, NOTICE HOW WE HAVE
THREE FUNCTION RULES, SO OUR GRAPH WILL HAVE THREE PIECES.
THE FIRST FUNCTION RULE IS F OF X = -2 IF X < -3.
THE SECOND FUNCTION RULE IS F OF X = 1 IF X = -3 AND < 1,
AND THE THIRD FUNCTION RULE IS F OF X = 4
IF X > OR = 1 AND < 4.
THIS IS A NICE EXAMPLE TO BEGIN WITH,
BECAUSE NOTICE HOW ALL THE FUNCTION RULES
ARE CONSTANT FUNCTIONS, OR FUNCTIONS IN THE FORM
OF F OF X = A CONSTANT, C, OR IF WE WANT, Y = C,
WHICH MEANS EACH PIECE WILL BE A PIECE OF A HORIZONTAL LINE.
SO NORMALLY THE WAY I LIKE TO DO THIS IS GO AHEAD AND GRAPH
THE ENTIRE FUNCTION, AND THEN DETERMINE WHAT PIECE
WE WANT BASED UPON THE INTERVAL FOR X.
SO FOR THE BLUE PIECE WE'RE GOING TO GRAPH
F OF X = -2, OR THE HORIZONTAL LINE Y = -2, AND THEN DETERMINE
WHAT PIECE WE WANT, OR THE PIECE WHEN X < -3.
SO F OF X = -2 IS THE HORIZONTAL LINE
PASSING THROUGH -2 ON THE Y AXIS,
OR THIS HORIZONTAL LINE HERE.
BUT AGAIN, WE ONLY WANT THE PIECE WHEN X < -3.
NOTICE HOW -3 IS NOT INCLUDED IN THIS INTERVAL.
SO WE'RE GOING TO GO TO -3 ON THE X AXIS,
GO DOWN TO OUR FUNCTION, BECAUSE WE DON'T INCLUDE -3,
WE'RE GOING TO MAKE AN OPEN POINT
OR AN OPEN CIRCLE HERE.
AND THEN X < -3 TO THE LEFT, SO WE ONLY WANT THIS BLUE PIECE
OF OUR HORIZONTAL LINE.
NOW THAT WE KNOW WHICH PIECE WE WANT, WE NEED TO ERASE
THE REST OF THE HORIZONTAL LINE.
NEXT WE'LL GRAPH THE FUNCTION F OF X = 1, AND THEN DETERMINE
WHICH PIECE WE WANT BASED UPON THIS INTERVAL.
SO WE'LL GRAPH THE HORIZONTAL LINE F OF X = 1,
WHICH WOULD BE THIS HORIZONTAL LINE HERE,
BUT WE ONLY WANT THE PIECE WHEN X = -3 AND < 1.
SO AGAIN, HERE'S -3.
NOTICE HOW -3 IS INCLUDED IN THIS INTERVAL,
SO WE'LL MAKE A CLOSED POINT AT -3 ON THIS HORIZONTAL LINE,
AND THEN WE'LL MOVE TO THE RIGHT TOWARD 1.
BUT NOTICE HOW 1 IS NOT INCLUDED IN THIS INTERVAL,
SO WE'LL MAKE AN OPEN CIRCLE OR OPEN POINT AT 1.
SO WE ONLY WANT THE BLUE PIECE OF THIS HORIZONTAL LINE,
SO WE NEED TO ERASE THE REST OF THE LINE.
AND THEN FOR THE LAST PIECE WE'LL GRAPH F OF X = 4,
AND THEN FIND THE PIECE THAT WE WANT.
SO HERE'S F OF X = 4.
BUT THIS IS THE FUNCTION RULE ONLY WHEN X > OR = 1 AND < 4.
SO HERE IS X = 1.
1 IS INCLUDED IN THIS INTERVAL, SO MAKE A CLOSED POINT HERE.
NOTICE HOW 4 IS NOT INCLUDED, SO MAKE AN OPEN POINT ON 4.
WE WANT THE PIECE BETWEEN THESE TWO, SO THE BLUE PIECE
IS THE GRAPH THAT WE WANT, SO WE ERASE THE REST.
NOW I REALIZE BECAUSE ALL OF THESE WERE HORIZONTAL
LINES WE PROBABLY DIDN'T HAVE TO GRAPH THE ENTIRE HORIZONTAL
LINE EACH TIME, BUT I THINK IT'S A GOOD TECHNIQUE
TO USE WHEN THESE FUNCTION RULES GET MORE INVOLVED.
I HOPE YOU FOUND THIS HELPFUL.