- WELCOME BACK TO A SECOND EXAMPLE OF DETERMINING
THE FUNCTION RULE FOR A PIECEWISE DEFINED FUNCTION
GIVEN THE GRAPH.
THE FIRST THING WE SHOULD RECOGNIZE FROM OUR GRAPH
IS THAT WE HAVE THREE PIECES, SO WE'LL HAVE TO HAVE
THREE FUNCTION RULES FOR OUR PIECEWISE DEFINED FUNCTION.
WORKING FROM LEFT TO RIGHT, WE'LL HAVE A FUNCTION RULE
FOR THIS PIECE.
IT'LL BE OUR FIRST FUNCTION RULE.
WELL, THE FUNCTION RULE FOR THIS PIECE,
OUR SECOND FUNCTION RULE, AND ON THE RIGHT
WE HAVE THIS PIECE HERE, WHICH WILL ALSO HAVE
ITS OWN FUNCTION RULE.
AND THEN FOR EACH PIECE WE'LL HAVE TO FIND THE DOMAIN
OR THE INTERVAL FOR X.
THIS ONE'S A LITTLE MORE CHALLENGING
THAN THE PREVIOUS EXAMPLE BECAUSE WE'LL HAVE TO DETERMINE
THE EQUATION OF THE LINE CONTAINING THE BLUE SEGMENT,
AND THE EQUATION OF THE LINE CONTAINING THE YELLOW SEGMENT,
WHICH WE'LL FIND BY USING SLOPE INTERCEPT FORM OF A LINE
WITH A FORM F OF X = MX + B WHERE M IS THE SLOPE
AND B IS THE Y INTERCEPT.
SO IF WE FOCUS ON THIS BLUE SEGMENT USING THE TWO ENDPOINTS,
WE SHOULD BE ABLE TO DETERMINE THE SLOPE.
MOVING FROM LEFT TO RIGHT FROM THIS POINT TO THIS POINT,
NOTICE HOW WE'D HAVE TO MOVE DOWN TWO UNITS,
SO THE CHANGE IN Y IS -2, AND THEN RIGHT TWO UNITS
MEANING THE CHANGE OF X IS +2, WHICH MEANS THE SLOPE
OR M IS GOING TO BE = TO -2 DIVIDED BY +2 OR -1.
NOW WE WANT TO DETERMINE THE Y INTERCEPT OF THE LINE
THAT WOULD CONTAIN THE SEGMENT.
WELL, IF WE WERE TO EXTEND THIS SEGMENT,
WE SHOULD BE ABLE TO SEE THAT THE LINE WOULD INTERSECT
THE Y AXIS AT +1, WHICH MEANS OUR Y INTERCEPT OR B IS +1.
ALSO KNOWING THE SLOPE IS -1, WHICH WE CAN WRITE AS -1/1,
FROM THIS POINT WE COULD GO DOWN ONE, RIGHT ONE, AND THEN AGAIN
DOWN ONE, RIGHT ONE TO DETERMINE OUR Y INTERCEPT IS +1.
SO IF M IS -1 AND B IS +1 OUR FUNCTION RULE
WOULD BE F OF X = -1 x X OR -X + 1.
THIS IS OUR FIRST FUNCTION RULE.
BUT WE ONLY WANT THIS LINE ON THE INTERVAL FROM -4 TO -2,
AND BECAUSE BOTH END POINTS ARE CLOSED POINTS,
WE WANT TO INCLUDE THE END POINTS.
WHICH MEANS X IS GREATER THAN OR = TO -4 AND LESS THAN
OR = TO -2.
THIS WOULD BE THE FUNCTION RULE AND THE INTERVAL
FOR THIS FIRST PIECE.
NEXT, THE GREEN PIECE IS A HORIZONTAL LINE
OR A CONSTANT FUNCTION, SO THAT FUNCTION RULE
WILL BE IN THE FORM OF F OF X = A CONSTANT.
IN THIS CASE WE'LL HAVE F OF X = +1 OR Y = +1.
SO THAT'S OUR FUNCTION RULE.
AND THEN FOR THE INTERVAL, IT'LL BE FROM -2 TO +1.
BECAUSE OF THE OPEN POINT, IT WILL NOT INCLUDE -2,
BUT BECAUSE OF THE CLOSED POINT HERE IT WILL INCLUDE +1.
SO X IS GREATER THAN -2 AND LESS THAN OR = TO +1.
AND NOW FOR THE LAST PIECE OR THE YELLOW PIECE, AGAIN,
WE'LL HAVE TO FIND THE EQUATION OF THE LINE
THAT WOULD CONTAIN THIS SEGMENT.
SO, AGAIN, WE'LL HAVE TO DETERMINE
THE SLOPE AND THE Y INTERCEPT OF THE LINE
CONTAINING THIS POINT.
TO DETERMINE THE SLOPE OF THIS LINE,
WE SHOULD BE ABLE TO USE ANY TWO CONVENIENT POINTS
ON THIS SEGMENT.
SO, FOR EXAMPLE, IF WE USE THIS POINT HERE
AND LET'S SAY THIS POINT HERE, AGAIN, WORKING
FROM LEFT TO RIGHT, NOTICE HOW WE'D HAVE A VERTICAL CHANGE
OR A CHANGE IN Y OF +2, AND A HORIZONTAL CHANGE OF +1
BECAUSE WE'RE MOVING RIGHT.
SO THE SLOPE OF THIS SEGMENT, OR THE LINE CONTAINING
THIS SEGMENT, WOULD BE +2.
SO THE SLOPE IS = TO +2.
NOW LET'S SEE IF THEY CAN DETERMINE THE Y INTERCEPT
OF THIS LINE.
WELL, IF WE EXTEND THIS SEGMENT, NOTICE HOW IT WOULD INTERSECT
THE Y AXIS AT +1, THEREFORE THE Y INTERCEPT IS +1.
ANOTHER WAY TO DETERMINE THIS Y INTERCEPT,
ALSO NOTICE FROM +1 ON THE Y AXIS,
IF WE GO UP TWO AND WRITE ONE, WE WOULD STILL BE ON THE LINE
CONTAINING THIS SEGMENT.
SO THE FUNCTION RULE WOULD BE F OF X = 2X + 1,
AND THE INTERVAL WOULD BE FROM +1 TO +4.
BECAUSE OF THE OPEN POINT, IT DOES NOT INCLUDE +1,
BUT BECAUSE OF THE CLOSED POINT, IT DOES INCLUDE +4.
SO WE WOULD SAY X IS GREATER THAN +1
AND LESS THAN OR = TO +4.
NOW, NOTICE FOR THE BLUE PIECE AND THE YELLOW PIECE,
WE WERE GRAPHICALLY ABLE TO DETERMINE THE EQUATIONS
OF THE LINES THAT WOULD CONTAIN THESE SEGMENTS.
BUT SOMETIMES WE MAY NOT BE ABLE TO DO THAT, SO I WANT TO SHOW
A SECOND WAY TO DETERMINE THE FUNCTION RULES
FOR THE BLUE PIECE AND THE YELLOW PIECE.
AND THIS WILL INVOLVE USING THE END POINTS OF THE SEGMENTS,
WHICH WOULD BE TWO POINTS ON THE LINE CONTAINING
THE TWO SEGMENTS.
SO FOR THIS BLUE SEGMENT, IF WE WANTED TO, WE COULD USE
THE END POINTS TO DETERMINE THE SLOPE OF THE LINE
USING OUR SLOPE FORMULA HERE.
SO FOR THE BLUE PIECE, USING OUR SLOPE FORMULA,
WE WOULD HAVE M = FOR Y2 - Y1 WE'D HAVE 3 - 5.
AND THEN FOR X2 - X1 WE'D HAVE -2 - -4
AND THAT WOULD BE -2/--THIS WOULD BE -2 + 4 OR +2,
WHICH = -1.
AND THEN WE CAN USE THE POINT SLOPE FORM OF THE LINE
TO DETERMINE OUR EQUATION.
SO WE'D HAVE Y--LET'S GO AHEAD AND USE THIS POINT HERE.
SO WE'D HAVE Y - 5 = -1 x X - 4 OR +4.
CLEAR THE PARENTHESIS AND SOLVE FOR Y.
WE'D HAVE Y - 5 = -1X - 4. ADD 5 TO BOTH SIDES,
YOU WOULD HAVE Y = OF COURSE THIS IS -X + 1.
NOTICE HOW IF WE GO BACK TO THE PREVIOUS SLIDE,
THIS IS THE SAME FUNCTION WHERE WE HAVE HERE FOR THE BLUE PIECE.
LET'S GO AHEAD AND DO THE SAME FOR THE YELLOW PIECE.
AGAIN, WE'LL USE THE END POINTS, WHICH WOULD BE TWO POINTS
ON THE LINE CONTAINING THIS SEGMENT.
SO TO FIND THE SLOPE FOR (Y2,- 1) WE'D HAVE (9,-3)
ALL OVER (X2,-X1) WOULD BE (4,- 1).
SO WE'D HAVE 6 DIVIDED BY 3, WHICH IS = TO 2.
AND THEN USING POINT SLOPE FORM OF A LINE, LET'S GO AHEAD
AND USE THIS FIRST POINT.
WE'D HAVE (Y - 3 = 2 x X - 1), CLEAR THE PARENTHESIS
AND THEN ADD 3 TO BOTH SIDES.
ADD 3 TO BOTH SIDES, WE HAVE Y = 2X + 1,
WHICH, AGAIN, IF WE GO BACK TO THE PREVIOUS SLIDE
WITH THE SAME FUNCTION WHERE WE FOUND GRAPHICALLY.
OKAY. THAT'S GOING TO DO IT FOR THIS EXAMPLE.
I HOPE YOU FOUND THIS HELPFUL.