- WE WILL BE INTERPRETING AND GRAPHING
PIECEWISE FUNCTIONS.
NOW YOU MAY THINK TO YOURSELF, "THERE ARE NO REAL LIFE
APPLICATIONS FOR PIECEWISE FUNCTIONS."
BUT IN FACT, IF YOU OWN A CELL PHONE
OR IF YOU'VE EVER SHIPPED A PACKAGE THROUGH UPS
OR THROUGH THE U.S. POSTAL SERVICE, THOSE RATES
ARE DETERMINED BY PIECEWISE FUNCTIONS.
SO THAT'S PRETTY EXCITING STUFF.
LET'S GO AHEAD AND JUMP INTO AN EXAMPLE.
IN 2005 SPRINT OFFERED A CELL PHONE CALLING PLAN
IN WHICH A CUSTOMER'S MONTHLY BILL CAN BE MODELED
BY THE GRAPH BELOW.
THE AMOUNT OF THE BILL IS A FUNCTION F OF THE NUMBER
OF MINUTES OF PHONE USE.
SO DOWN HERE ON THE X AXIS WE HAVE THE NUMBER OF MINUTES
USED FOR A MONTH, AND ON THE Y AXIS
WE HAVE THE MONTHLY BILL IN DOLLARS.
UNDER THIS PLAN, IF THE CUSTOMER USES THE PHONE
FOR 360 MINUTES, WHAT IS HIS OR HER MONTHLY BILL?
I'LL GO BACK TO THE LARGER GRAPH ON THE PREVIOUS SCREEN.
LOCATE 360 ON THE X AXIS.
GO STRAIGHT TO THE GRAPH.
THE GRAPH IS DEFINED HERE WHERE Y = 40,
SO THE BILL WOULD BE $40.
I WOULD LIKE TO GO BACK AND LOOK AT ANOTHER CASE
BEFORE I MOVE ON TO THE NEXT QUESTION.
WHAT IF, FOR EXAMPLE, YOU USE 700 MINUTES?
AS YOU GO UP TO THE GRAPH, NOTICE HOW THERE'S A CLOSED
POINT HERE AND THERE'S AN OPEN POINT HERE.
WELL, WHERE IS THE FUNCTION DEFINED?
REMEMBER THE FUNCTION IS DEFINED AT THE CLOSED POINT,
WHICH WOULD RESULT IN A Y VALUE OF $55.
THE FUNCTION IS NOT DEFINED AT THE OPENED POINTS,
JUST TO REVIEW.
THE NEXT QUESTION, IF A MONTHLY BILL IS $55,
FOR HOW MANY MINUTES DID THE CUSTOMER USE THE PHONE?
WELL, WE JUST KIND OF STARTED ANALYZING THAT.
IF THE PERSON PAID $55, THEY COULD'VE USED--NOTICE
THERE'S A HORIZONTAL INTERVAL WHERE THE BILL IS $55.
IN FACT, IF A CUSTOMER USED LET'S SAY 640 MINUTES,
THE BILL WOULD BE $55.
620 MINUTES WOULD STILL BE A BILL OF $55.
NOW, 600 IS KIND OF AT THE DIVIDING LINE.
IF THE CUSTOMER USED 600 MINUTES,
NOW THE BILL WOULD BE $50.
BUT IF THEY USED 601 MINUTES, IT WOULD JUMP UP TO $55.
IN FACT, IF IT WAS 600.001 MINUTES IT WOULD JUMP
TO THE NEXT INTERVAL.
SO HOW DO WE EXPRESS THAT?
LET'S GO BACK TO THE QUESTION.
IF WE LET T = NUMBER OF MINUTES USED, WE KNOW
THAT IT HAS TO BE > OR = 700.
BUT T ALSO HAS TO HAVE A LOWER BOUND, AND IT HAS TO BE < 600.
NOTICE HOW THIS INEQUALITY SYMBOL DOES NOT INCLUDE
THE VALUE OF 600.
OKAY. SO THE INTERVAL FROM 600 TO 700, INCLUDING 700,
NOT INCLUDING 600.
LET'S TAKE A LOOK AT THE GRAPH OF PIECEWISE FUNCTIONS.
FOR EXAMPLE, GRAPH THE FUNCTION DEFINED AS FOLLOWS:
G OF X = 2 FOR X > OR = 1, G OF X = -X - 2 FOR X < 1.
THERE ARE TWO PIECES TO THIS FUNCTION,
SO THE END RESULT MAY HAVE TWO PIECES.
AND HERE'S HOW WE WANT TO INTERPRET THIS.
G OF X = 2, AND THAT'S THE HORIZONTAL LINE, Y = 2,
BUT ONLY FOR X > OR = 1.
SO X IS > OR = 1 STARTING AT 1, AND WOULD READ TO THE RIGHT.
NOW THE SECOND PIECE OF THIS FUNCTION
IS THE LINE G OF X = -X - 2 ONLY FOR X < 1.
SO REMEMBER THIS LINE IS Y INTERCEPT OF -2
SLOPE OF -1.
IT WOULD BE THIS RED LINE.
BUT NOTICE HOW WE STOPPED IT WHEN X = 1, AND SINCE IT
DID NOT INCLUDE 1 WE MADE AN OPEN CIRCLE TO SHOW THAT,
AND THEN ONLY THE GRAPH ON THE LEFT SIDE OF X = 1.
NOW THESE TWO GRAPHS MAKE UP THE FUNCTION G OF X,
SO WE CAN'T KEEP THEM SEPARATE.
WE HAVE TO COMBINE THEM ONTO THE SAME COORDINATE PLANE
TO LOOK LIKE THIS.
OKAY.
HOPEFULLY THAT EXPLAINS THIS PIECEWISE FUNCTION.
LET'S TAKE A LOOK AT ONE MORE.
NOW THIS ONE LOOKS LIKE IT MAY HAVE THREE PIECES,
AND IN FACT IT DOES.
AND I'M GOING TO SHOW THIS ONE ALSO ON THE GRAPHING CALCULATOR,
BUT LET'S KIND OF EXPLAIN THIS INTERVAL BY INTERVAL.
THE FIRST INTERVAL WE HAVE THE HORIZONTAL LINE F OF X = 3
WHEN X < OR = 0.
HERE IS THE HORIZONTAL LINE Y = 3 ONLY WHEN X < 0.
THE NEXT PIECE.
WE WANT THE FUNCTION F OF X = 1 - X SQUARED.
REMEMBER, THAT'S A PARABOLA, BUT WE ONLY WANT A SMALL PIECE
OF IT FROM 0 TO 2, NOT INCLUDING 0,
BUT INCLUDING 2.
SO NOTICE HOW WE HAVE AN OPEN CIRCLE WHEN X IS 0,
AND WE COULD PUT A CLOSED POINT
IF WE WANTED TO RIGHT THERE WHERE X IS 2.
I THINK I'LL DO THAT JUST TO EMPHASIZE THAT.
AND THE LAST PIECE WE HAVE THE LINE F OF X = 2X - 4,
THIS BLACK LINE.
WE WANT THAT BLACK LINE WHEN X > 2, NOT INCLUDING 2.
SO WE HAVE AN OPEN CIRCLE AT X = 2, AND THEN >
MEANING GOING TO THE RIGHT.
WE HAVE THREE PIECES.
NOW, IF WE WANT TO GRAPH THIS ON OUR GRAPHING CALCULATOR
WE CAN.
WE HAVE TO ENTER IT IN THREE DIFFERENT TIMES.
LET'S TAKE A LOOK AT THAT.
THE WAY WE DO THIS IS WE GO TO OUR Y = SCREEN.
OKAY. LET'S TAKE A LOOK AT HOW WE WOULD GRAPH THIS
ON THE GRAPHING CALCULATOR.
IT IS KIND OF INVOLVED ON THE GRAPHING CALCULATOR
AS WELL.
IT IS BY NO MEANS AN EASY PROCESS.
AND THE GENERAL PROCEDURE IS THIS:
GO AHEAD AND HIT Y = WHERE YOU'RE ON THE SCREEN
THAT I'M ON.
FIRST YOU TYPE IN YOUR GIVEN RULE OR FUNCTION FOR EACH PIECE,
AND THEN YOU MULTIPLY IT BY IN PARENTHESES
THE INTERVAL FOR WHICH YOU WANT TO CONSIDER THAT GRAPH.
SO FOR EXAMPLE, THE FIRST FUNCTION WAS F OF X = 3,
SO I TYPED IN THE 3, AND THEN I MULTIPLIED IT
BY X < OR = 0.
NOW THE SECOND PIECE OF THE FUNCTION,
1 - X SQUARED, SINCE I HAVE THE SUM OR DIFFERENCE
OF TWO TERMS I DO HAVE TO PUT THAT FUNCTION RULE IN A SET
OF PARENTHESES.
AND I MULTIPLIED IT BY--IN THIS CASE I HAVE TWO INTERVALS
TO MULTIPLY IT BY.
YOU CANNOT TYPE THIS IN AS A COMPOUND INTERVAL
OR A COMPOUND INEQUALITY AND CALCULATE IT,
SO I HAD TO TYPE IN 0 < X IN PARENTHESES,
AND X > OR = 2.
I'LL SHOW YOU HOW TO TYPE IN THOSE INEQUALITIES
IN JUST A MOMENT.
AND THE THIRD PIECE, THE LINE 2X - 4 IN PARENTHESES,
AND I MULTIPLIED THAT BY X > 2.
NOW IN ORDER TO FIND THESE INEQUALITY SYMBOLS
IT'S A PRETTY EASY PROCESS. JUST HIT SECOND MATH,
WHICH BRINGS UP THE TEST MENU, AND YOU PICK THOSE OPTIONS
THAT YOU NEED.
NOW, I'M NOT GOING TO GO THROUGH TYPING EACH ONE OF THESE IN,
DUE TO TIME, BUT HOPEFULLY THAT WILL GIVE YOU A GOOD IDEA
OF HOW YOU CAN DO THIS.
ANOTHER DRAWBACK TO THE GRAPHING CALCULATOR
IS WHEN YOU DO GRAPH IT, IT DOES NOT INCLUDE
THE OPEN AND CLOSE POINTS.
YOU STILL HAVE TO DETERMINE THAT.