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- WELCOME TO INTRODUCTION TO FUNCTIONS.
THE GOALS OF THIS VIDEO ARE TO DEFINE A FUNCTION
AND ALSO TO IDENTIFY THE DOMAIN AND RANGE
OF A FUNCTION.
A FUNCTION IS JUST A RULE THAT FOR EVERY INPUT
ASSIGNS A SPECIFIC OUTPUT.
YOU CAN ALSO THINK OF A FUNCTION AS A MACHINE IN WHICH EACH INPUT
PRODUCES A SINGLE OUTPUT.
AS AN EXAMPLE, LET'S SAY YOU OWN A PRE-PAID CELL PHONE.
THE MONTHLY COST IS A FUNCTION OF THE NUMBER OF MINUTES
THAT YOU USE.
FOR EXAMPLE, IF THE COST IS $.15 PER MINUTE;
YOUR MONTHLY COST WILL BE THE NUMBER OF MINUTES
TIMES $.15.
SO IF WE THINK OF THIS BOX AS THE FUNCTION THAT REPRESENTS
THIS SITUATION; THE INPUT WOULD BE MULTIPLIED BY $.15.
SO FOR EXAMPLE, IF YOU USED 100 MINUTES ONE MONTH,
100 x $.15 WOULD BE $15.
SO THE INPUT WOULD BE 100 MINUTES AND THE OUTPUT
WOULD BE $15.
IF THE NEXT MONTH YOU USED 200 MINUTES THEN 200 x $.15
WOULD BE $30.
SO AN INPUT OF 200 MINUTES HAS AN OUTPUT OF $30 AND SO ON.
SO WE CAN SEE THAT FOR EVERY INPUT
THERE'S ONLY ONE OUTPUT.
TO THINK OF THIS IN A MORE MATHEMATICAL WAY
WE COULD DEFINE 2 VARIABLES WHERE 1 WOULD REPRESENT
THE NUMBER OF MINUTES AND 1 WOULD REPRESENT
THE MONTHLY COST.
INPUT IS USUALLY X SO WE'LL LET X = THE NUMBER OF MINUTES
AND WE'LL LET Y = THE MONTHLY COST.
SO IF WE WANTED TO WRITE AN EQUATION TO REPRESENT
THIS SITUATION WE COULD SEE THAT THE MONTHLY COST Y
IS = TO 0.15 x X.
HOWEVER, WHEN WORKING WITH FUNCTIONS WE USUALLY USE
SOMETHING CALLED FUNCTION NOTATION;
WHICH MEANS INSTEAD OF WRITING Y = 0.15X WE CAN WRITE
F OF X = 0.15X.
SO F OF X IS = TO Y AND THIS NOTATION HERE
IS USED TO SHOW THE RELATIONSHIP BETWEEN THE INPUT AND THE OUTPUT
IS A FUNCTION.
SO WE'RE RELATING THIS NOTATION TO WHAT WE WERE SAYING EARLIER
ABOUT THE INPUT IS MULTIPLIED BY $.15
TO PRODUCE THE OUTPUT.
WE COULD SAY THAT F OF 100, MEANING 100 IS THE INPUT
INTO THE FUNCTION AND THE OUTPUT WOULD BE $15.
AND F OF 200 IS = TO $30.
ONE MORE CONNECTION WE SHOULD MAKE HERE
IS THAT EACH OF THESE FUNCTION VALUES REPRESENTS
A POINT ON THE GRAPH OF THE FUNCTION.
SO F OF 100 = $15 WOULD REPRESENT THE POINT
WHERE X IS 100 AND Y IS 15.
AND F OF 200 = 30 REPRESENTS THE POINT WHERE X IS 200
AND THE Y-COORDINATE WOULD BE 30.
SO THERE'S A LOT GOING ON HERE,
BUT OTHER THAN SOME DIFFERENT NOTATION
IT IS THINGS THAT WE'VE ALREADY STUDIED BEFORE
WHEN WE HAD EQUATIONS IN TERMS OF X AND Y.
AND THERE IS QUITE A BIT OF NEW VOCABULARY
WHEN WE TALK ABOUT FUNCTIONS.
THE INPUT, USUALLY X,
IS CALLED THE INDEPENDENT VARIABLE.
AND THE OUTPUT, USUALLY Y,
IS CALLED THE DEPENDENT VARIABLE.
LET'S TAKE A LOOK AT ANOTHER SITUATION
THAT MODELS A FUNCTION.
LET'S SAY THAT YOU PAY $70 PER MONTH
FOR DIGITAL CABLE TELEVISION AND THEN YOUR CABLE COMPANY
OFFERS PAY PER VIEW MOVIES FOR $3.00 PER MOVIE.
LET'S SAY WE WANT TO WRITE A FUNCTION THAT MODELS
THE TOTAL COST OF YOUR CABLE BILL.
WELL, NOW THERE'S TWO COMPONENTS TO YOUR CABLE BILL.
LET'S SAY THIS FIRST COMPONENT WOULD BE THE FIXED COST
OR THE MONTHLY COST OF $70 AND THEN THE SECOND BOX
REPRESENTS THE VARIABLE COST OR THE COST THAT VARIES
BASED UPON THE NUMBER OF MOVIES THAT YOU WATCH.
AND THE VARIABLE COST WOULD BE THE NUMBER OF MOVIES
THAT YOU WATCH x $3.00.
SO FOR THIS SITUATION THE INPUT WOULD BE THE NUMBER OF MOVIES
THAT YOU WATCH THAT MONTH AND THE OUTPUT WOULD BE
THE TOTAL COST FOR THAT MONTH.
SO FOR EXAMPLE IF YOU WATCHED 10 MOVIES IN ONE MONTH
YOU'D HAVE TO PAY $70 + 10 x $3.00 OR $30;
WHICH WOULD GIVE US A TOTAL COST OF $100 FOR THAT MONTH.
IF YOU WATCH 20 MOVIES IN ONE MONTH YOU'D HAVE
$70 + 20 x $3.00; WHICH WOULD BE $60.
$70 + $60 WOULD = $130.
TO WRITE AN EQUATION AND REPRESENT THIS SITUATION
WE COULD LET X = THE NUMBER OF MOVIES WATCHED
AND Y WOULD BE THE TOTAL OF MONTHLY COST.
SO THE EQUATION IN TERMS OF X AND Y WOULD BE
Y THE TOTAL COST WOULD = $70 + THE NUMBER
OF MOVIES x $3.00 OR 3 x X.
BUT WE KNOW THIS RELATIONSHIP IS A FUNCTION SO WE CAN REPLACE
Y WITH F OF X.
SO USING FUNCTION NOTATION WE WOULD WRITE THIS AS
F OF X = 70 + 3X.
AND JUST TO ILLUSTRATE THE CONNECTION ONCE AGAIN,
IF WE WATCH 10 MOVIES IN ONE MONTH WE WOULD SAY
F OF 10 WOULD = 70 + 3 x 10 WHICH WOULD BE $100
SO F OF 10 = 100.
NOW, LET'S TAKE A LOOK AT SOME ADDITIONAL VOCABULARY
WHEN DEALING WITH FUNCTIONS.
THE SET OF ALL POSSIBLE INPUTS IS CALLED THE DOMAIN
OF THE FUNCTION.
SO THE DOMAIN CAN BE VIEWED AS A SET OF ALL POSSIBLE X VALUES
AND THE SET OF ALL POSSIBLE OUTPUTS IS CALLED THE RANGE.
SO WE CAN THINK OF THE RANGE AS THE SET
OF ALL POSSIBLE Y VALUES.
LET'S SEE IF WE CAN DETERMINE THE DOMAIN AND RANGE
FOR THIS SITUATION.
THE COST OF A TAXI IS A $4.00 FLAG FEE + $3.00 PER MILE.
SO THE $4.00 FLAG FEE IS WHAT YOU'RE CHARGED
AS SOON AS YOU ENTER INTO THE TAXI.
SO THAT'S THE FLAT RATE AND THEN YOU HAVE TO PAY $3.00
PER MILE AFTER THAT.
SO IF YOU TOOK A TAXI FOR ONE MILE YOU'D HAVE
$4.00 + A $3.00 FEE OR A TOTAL OF $7.00.
IF YOU WENT FOR TWO MILES YOU'D HAVE
THE $4.00 FEE + 2 x $3.00 WHICH WOULD BE $6.00
FOR A TOTAL OF $10.
SO IF WE LET X = THE NUMBER OF MILES AND Y = THE TOTAL COST
WE WOULD HAVE Y = THE $4.00 FEE + $3.00 x X
OR THE NUMBER OF MILES.
SO USING FUNCTION NOTATION WE'D HAVE F OF X = 4 + 3X.
SO NOW WHEN WE TALK ABOUT THE DOMAIN AND THE RANGE
WE'RE TALKING ABOUT WHAT ARE THE POSSIBLE VALUES
FOR X AND X REPRESENTS THE NUMBER OF MILES
A TAXI WOULD DRIVE.
SO WE COULD GET IN THE TAXI AND THEN GET RIGHT OUT
AND WE'D STILL HAVE A $4.00 FEE.
SO IT COULD = 0, BUT X COULD NEVER BE NEGATIVE
BECAUSE WE CAN'T DRIVE NEGATIVE MILES.
SO THE DOMAIN WOULD BE ANY NUMBER THAT IS > OR = TO 0.
NOW, THE RANGE ARE ALL THE POSSIBLE Y VALUES.
WELL LOOKING AT THIS COST EQUATION; IF WE GO 0 MILES
WE STILL HAVE THIS $4.00 FLAG FEE FOR CALLING THE TAXI.
SO THE COST OF THE TAXI IS GOING TO BE $4.00 OR MORE.
SO WE COULD SAY THAT THE RANGE WOULD BE
Y > OR = TO 4.
LET'S TAKE A LOOK AT ANOTHER EXAMPLE
OF DETERMINING THE DOMAIN AND RANGE FROM A SITUATION
THAT REPRESENTS A FUNCTION.
HERE IT COSTS $5.00 TO ENTER THE THEME PARK
AND $2.00 FOR EACH RIDE.
SO IF WE LET X = THE NUMBER OF RIDES
AND AGAIN Y = THE TOTAL COST WE WOULD HAVE Y = THE $5.00
COST TO ENTER THE PARK + $2.00 x THE NUMBER
OF RIDES OR 2X.
SO USING FUNCTION NOTATION WE REPLACE Y WITH F OF X,
SO WE HAVE F OF X = 5 + 2X.
SO TO DETERMINE THE DOMAIN AND RANGE FOR THIS SITUATION
LET'S GO AHEAD AND MAKE A T-TABLE.
REMEMBER X REPRESENTS THE NUMBER OF RIDES,
SO X COULD BE 0, 1, 2, 3 AND SO ON.
LET'S SEE WHAT Y WOULD BE FOR THESE DIFFERENT X VALUES.
WELL, WHEN X IS 0 WE'D HAVE 5 + 2 x 0 THAT'S 5.
WHEN X IS 1 WE'D HAVE 5 + 2 x 1 THAT'S 7.
WHEN X IS 2 THIS WOULD BE 4 SO 5 + 4 THAT'D BE 9.
AND THEN WHEN X IS 3 WE'D HAVE 2 + THIS WOULD BE 6
OR 11 AND SO ON.
SO WE NEED TO THINK ABOUT HOW ARE WE GOING TO EXPRESS
THE DOMAIN AND THE RANGE BASED UPON THESE GIVEN X VALUES.
NOTICE THAT X COULD NOT BE A DECIMAL OR FRACTION
BECAUSE WE CAN'T GO ON A DECIMAL
OR FRACTION NUMBER OF RIDES.
SO THE DOMAIN WOULD BE THE SET OF NUMBERS 0, 1, 2, 3 AND SO ON.
ONE WAY TO SAY THIS WOULD BE TO STATE THE DOMAIN
IS ALL NON-NEGATIVE INTEGERS.
AND NOW LETS TAKE A LOOK AT THE RANGE.
THE RANGE ARE THE NUMBERS 5, 7, 9, 11 AND SO ON.
SO IF WE WERE TO LIST THESE IT WOULD LOOK LIKE THIS.
NOW, IN REALITY THERE IS A MAXIMUM NUMBER OF RIDES
SOMEBODY COULD GO ON IN ONE DAY;
THEREFORE THERE ARE SOME MAXIMUM COSTS,
BUT WE'RE NOT GIVEN ENOUGH INFORMATION IN THIS PROBLEM
SO WE'LL GO AHEAD AND ASSUME THIS COULD
CONTINUE ON FOREVER.
SO THE QUESTION BECOMES: HOW DO WE EXPRESS THESE VALUES
IN THE RANGE?
AND NOTICE THE PATTERN IS A SEQUENCE OF ODD NUMBERS.
SO WE COULD SAY THAT THE RANGE WOULD BE
ALL ODD NUMBERS THAT ARE > OR = TO 5.
OKAY, THAT'LL DO IT FOR THIS INTRODUCTORY VIDEO.
IN THE NEXT VIDEO WE'LL TAKE A LOOK AT
GRAPHING FUNCTIONS TO DETERMINE THE DOMAIN
AND RANGE.