- WELCOME TO ANOTHER EXAMPLE OF GRAPHING
A PIECEWISE DEFINED FUNCTION.
LOOKING AT OUR PIECEWISE FUNCTION,
NOTICE HOW WE HAVE THREE FUNCTION RULES,
SO GRAPHICALLY WE COULD EXPECT THREE PIECES.
F OF X = -X - 1 IF X > OR = -3 AND < 1, AND F OF X = -1
IF X > OR = 1 AND < 3.
NOTICE F OF X = 4 AND F OF X = -1
ARE BOTH CONSTANT FUNCTIONS, OR FUNCTIONS IN THE FORM
OF F OF X = CONSTANT C, OR IF WE WANT, Y = C,
WHICH MEANS THE GRAPHS OF THESE WILL BE PIECES
OF HORIZONTAL LINES.
AND THEN F OF X = -X - 1 IS A LINEAR FUNCTION
IN SLOPED INTERCEPT FORM, OR THE FORM F OF X = MX + B,
WHERE B IS THE Y INTERCEPT OF THE LINE AND M
IS THE SLOPE OF THE LINE, WHICH MEANS THIS PIECE
WILL BE A PIECE OF A SLANTED LINE.
NORMALLY WHAT I PREFER TO DO IS GRAPH THE ENTIRE FUNCTION,
AND THEN DETERMINE WHAT PIECE OF THE GRAPH WE WANT
BASED UPON THE INTERVAL FOR X.
SO WE'LL START BY GRAPHING F OF X = 4, WHICH WOULD BE
A HORIZONTAL LINE PASSING THROUGH 4 ON THE Y AXIS.
SO IT WOULD LOOK SOMETHING LIKE THIS.
BUT WE ONLY WANT THE PIECE OF THIS LINE WHEN X < -3,
SO WE'LL FIND -3 ON THIS LINE, WHICH WOULD BE RIGHT HERE.
NOTICE -3 IS NOT ON THIS INTERVAL,
SO WE MAKE AN OPEN POINT ON -3.
AND X < -3 TO THE LEFT, SO WE ONLY WANT THE BLUE PIECE
OF THIS HORIZONTAL LINE, WHICH MEANS WE NEED TO ERASE
THE REST OF THE LINE.
THIS GRAPH HERE IS THE GRAPH OF F OF X = 4 IF X < -3.
THE NEXT FUNCTION RULE F OF X = -X - 1
WILL BE THE GRAPH OF A LINE WHERE THE SLOPE = -1 OR -1/1,
AND THE Y INTERCEPT, OR B, = -1.
AGAIN WE'LL GRAPH THIS ENTIRE LINE,
AND THEN DETERMINE WHAT PIECE IS IN THE GIVEN INTERVAL.
SO WE'LL PLOT THE Y INTERCEPT OF -1, WHICH WOULD BE HERE.
THE SLOPE IS -1/1, WHICH MEANS WE'LL GO DOWN 1 AND RIGHT 1,
DOWN 1 AND RIGHT 1, DOWN 1 AND RIGHT 1, AND SO ON.
OUR LINEAR FUNCTION WILL PASS THROUGH THESE FOUR POINTS,
AND LOOK SOMETHING LIKE THIS.
BUT WE ONLY WANT THE PIECE OF THE LINE
WHEN X > OR = -3 AND < 1.
SO WE'LL FIND WHERE X = -3 ON THE LINE,
WHICH IS THIS POINT HERE.
-3 IS IN THE INTERVAL THIS TIME, SO WE'LL MAKE
A CLOSED POINT HERE, AND THEN WE'LL START
APPROACHING 1.
BUT NOTICE HOW 1 IS NOT IN THIS INTERVAL,
SO WE'LL HAVE AN OPEN POINT WHEN X = 1.
SO WE'LL MAKE AN OPEN POINT HERE.
WE ONLY WANT THE BLUE PIECE OF THIS LINE,
SO WE ERASE THE REST OF THE LINE.
AND THEN FOR THE LAST PIECE WE HAVE F OF X = -1,
WHICH WILL BE A HORIZONTAL LINE PASSING THROUGH
-1 ON THE Y AXIS.
SO IT WILL LOOK SOMETHING LIKE THIS.
BUT WE ONLY WANT THE PIECE WHEN X > OR = 1 AND < 3.
1 IS IN THIS INTERVAL, SO ON THIS LINE WHERE X = 1
WE HAVE A CLOSED POINT.
WE'LL APPROACH 3, BUT 3 IS NOT IN THE INTERVAL,
SO WE'LL HAVE AN OPEN POINT ON 3.
AND NOW WE NEED TO ERASE THE REST OF THE HORIZONTAL LINE,
LEAVING ONLY THE BLUE PIECE.
SO THIS WOULD BE THE GRAPH OF OUR PIECEWISE
DEFINED FUNCTION.
THIS PIECE CORRESPONDS TO THE BLUE FUNCTION RULE,
THIS PIECE CORRESPONDS TO THE GREEN FUNCTION RULE,
AND THIS PIECE CORRESPONDS TO THE YELLOW FUNCTION RULE.
I HOPE YOU FOUND THIS HELPFUL.