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- WELCOME TO AN EXAMPLE ON HOW TO DETERMINE
THE FUNCTION RULE FOR A PIECEWISE DEFINED FUNCTION
GIVEN THE GRAPH OF THE FUNCTION.
THE FIRST THING WE SHOULD NOTICE IS THE GRAPH HAS THREE PIECES,
THEREFORE OUR PIECEWISE DEFINED FUNCTION WILL HAVE
TO HAVE THREE SEPARATE FUNCTION RULES.
WE'LL HAVE A FUNCTION RULE FOR THIS PIECE,
WHICH WILL BE OUR FIRST FUNCTION RULE.
THEN WE'LL HAVE A FUNCTION RULE FOR THE SECOND PIECE,
WHICH WILL BE OUR SECOND FUNCTION RULE.
AND THEN FOR THIRD PIECE, WE'LL HAVE A THIRD
FUNCTION RULE.
THIS IS A NICE EXAMPLE TO BEGIN WITH BECAUSE EACH PIECE
IS A CONSTANT FUNCTION MEANING EACH PIECE
IS A HORIZONTAL SEGMENT.
REMEMBER ALL HORIZONTAL LINES ARE IN THE FORM
OF Y = SOME CONSTANT, OR IN THIS CASE
F OF X = A CONSTANT.
SO WE'LL DEFINE THE FUNCTION RULE FOR EACH PIECE,
AND THEN WE'LL DEFINE THE DOMAIN OR THE X VALUES FOR EACH PIECE.
SO IF WE START WITH THIS BLUE PIECE,
NOTICE THAT WE WOULD HAVE THE EQUATION Y = -3,
OR IN THIS CASE F OF X = -3.
BUT NOTICE THIS IS ONLY TRUE ONLY INTERVAL
BETWEEN -4 AND -1 WHERE IT DOES NOT INCLUDE -4,
BUT IT DOES INCLUDE -1.
SO F OF X = -3 IF X IS GREATER THAN -4 AND LESS THAN
OR = TO -1.
THIS WOULD BE THE DOMAIN OR THE INTERVAL
FOR X WHEN F OF X = -3.
NOW, LOOKING AT THE GREEN PIECE, NOTICE HOW WE'D HAVE
THE FUNCTION Y = -1 OR F OF X = -1.
AND THE INTERVAL FOR THIS PIECE IS FROM -1 TO +2.
NOTICE HOW IT DOES NOT INCLUDE EITHER END POINT.
THAT'S WHY WE HAVE OPEN POINTS ON EACH END.
SO THE INTERVAL IS GOING TO BE WHEN X IS GREATER THAN -1
AND LESS THAN +2.
THIS IS THE INTERVAL FOR WHICH F OF X = -1.
AND THEN FOR THE LAST PIECE OR THE YELLOW PIECE,
NOTICE HOW THIS WOULD BE THE FUNCTION Y = 2
OR F OF X = 2.
SO THERE'S OUR FUNCTION RULE, AND THIS IS A RULE
ONLY ON THE INTERVAL FROM +2 TO +4.
BUT IN THIS CASE WE DO INCLUDE BOTH END POINTS, SO THE INTERVAL
OR DOMAIN FOR THIS PIECE WOULD BE X IS GREATER THAN
OR = TO +2 AND LESS THAN OR = TO +4.
WE'LL TAKE A LOOK AT ANOTHER EXAMPLE IN THE NEXT VIDEO
WHERE THE FUNCTION RULES ARE NOT SO BASIC,
OR NOT JUST CONSTANT FUNCTIONS.
BUT THIS IS A NICE EXAMPLE TO BEGIN WITH TO PRACTICE
TO FINDING THE DOMAINS OR INTERVALS FOR X
TO REACH FUNCTION RULE.
I HOPE YOU FOUND THIS HELPFUL.