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X
- WE WANT TO FIND THE VALUE OF M
TO MAKE F OF X CONTINUOUS EVERYWHERE.
WELL NOTICE F OF X IS A PIECEWISE DEFINED FUNCTION
WHERE M WOULD BE THE SLOPE OF THE LINE WHEN X
AND WHEN X > -2 WE HAVE A QUADRATIC FUNCTION.
WELL WE KNOW THAT A LINEAR FUNCTION
AND A QUADRATIC FUNCTION BY THEMSELVES ARE ALWAYS CONTINUOUS
WHICH MEANS TO MAKE THIS PIECEWISE DEFINED FUNCTION
CONTINUOUS EVERYWHERE,
WE ONLY HAVE TO MAKE IT CONTINUOUS WHEN X = -2.
SO WHILE WE DO HAVE FORMAL RULES THAT DEFINE
WHEN A FUNCTION IS CONTINUOUS AT A POINT GIVEN HERE
I THINK IT WILL BE MORE HELPFUL
IF WE TAKE A LOOK AT THIS GRAPHICALLY.
REMEMBER IF A FUNCTION IS CONTINUOUS EVERYWHERE
WE SHOULD BE ABLE TO SKETCH IT WITHOUT LIFTING UP OUR PENCIL.
SO FOR EXAMPLE, IF THIS WAS OUR PIECEWISE DEFINED FUNCTION
THIS WOULD BE CONTINUOUS EVERYWHERE
BECAUSE NOTICE HOW WE CAN SKETCH THIS GRAPH
WITHOUT LIFTING UP OUR PENCIL.
BUT IF THIS WAS THE GRAPH OF OUR PIECEWISE DEFINED FUNCTION
NOTICE HOW THIS WOULD NOT BE CONTINUOUS AT X = -2
BECAUSE WE HAVE A BREAK IN THE GRAPH
AND WE'D HAVE TO LIFT UP OUR PENCIL
IN ORDER TO SKETCH THE TWO PIECES.
SO IF WE WANT OUR GRAPH TO LOOK LIKE THIS
THAT MEANS THAT WHEN X = -2
THE TWO FUNCTION RULES MUST EQUAL EACH OTHER
MEANING MX - 2 MUST = X SQUARED + 2X - 6 WHEN X = -2.
SO WE CAN SET THIS UP AS AN EQUATION
AND THEN SOLVE FOR M.
AGAIN, IN ORDER FOR THIS TO BE CONTINUOUS EVERYWHERE
MX - 2 MUST = X SQUARED + 2X - 6 WHEN X = -2.
SO FOR THE FIRST STEP, WE'LL SUBSTITUTE -2 FOR X.
SO M x -2 = -2M - 2 = -2 SQUARED = 4 + 2 x -2 = -4 - 6.
SO NOW WE'LL SOLVE FOR M.
WE HAVE -2M - 2 = -6. ADD 2 TO BOTH SIDES.
SO WE HAVE -2M = -4. DIVIDE BOTH SIDES BY -2.
SO WE HAVE M = +2 WHICH MEANS IN ORDER FOR THIS FUNCTION
TO BE CONTINUOUS EVERYWHERE
THE FIRST FUNCTION RULE WOULD BE F OF X = 2X - 2
IF X
NOW LET'S GO BACK AND TAKE A LOOK AT OUR GRAPH.
WHEN X
THE FUNCTION RULE FOR THIS LINE HERE WOULD BE F OF X = 2X - 2,
AND THEN WHEN X > -2
THE FUNCTION RULE IS F OF X = X SQUARED + 2X - 6 AS GIVEN.
SO AGAIN, WE FOUND THAT M MUST EQUAL +2
IN ORDER FOR THIS FUNCTION TO BE CONTINUOUS EVERYWHERE.
AND IF WE WANTED TO WE COULD NOW GO THROUGH EACH OF THESE RULES
FOR CONTINUITY AT A POINT WHEN C = -2
TO VERIFY THAT IT IS CONTINUOUS AT THAT POINT
BUT WE CAN TELL FROM THE GRAPH THAT IT IS.
OKAY. I HOPE YOU FOUND THIS EXAMPLE HELPFUL.