- THE IDEA OF A FUNCTION IS ONE OF THE MOST IMPORTANT CONCEPTS
IN MATHEMATICS.
BASICALLY A FUNCTION IS A SPECIAL RELATION
OR CORRESPONDENCE BETWEEN TWO SETS.
THE GOAL OF THIS VIDEO IS TO DETERMINE
IF A RELATION OR A CORRESPONDENCE IS A FUNCTION.
LET'S TAKE A LOOK AT THE DEFINITION OF A FUNCTION.
THE FORMAL DEFINITION:
A FUNCTION IS A CORRESPONDENCE BETWEEN A FIRST SET,
CALLED THE DOMAIN,
AND THE SECOND, SET CALLED THE RANGE,
SUCH THAT EACH MEMBER OF THE DOMAIN
CORRESPONDS TO EXACTLY ONE MEMBER OF THE RANGE.
I'D LIKE TO EXPLAIN THIS WITH A LITTLE MORE DETAIL.
WHEN YOU HEAR THE WORD DOMAIN YOU CAN BASICALLY THINK OF,
IN MOST CASES, THE SET OF X VALUES.
AND THE RANGE, YOU CAN THINK OF IN MOST CASES,
AS THE SET OF Y VALUES.
SO IF YOU HAVE A SET OF X AND Y VALUES,
IT WOULD BE A FUNCTION IF EVERY X VALUE IS PAIRED
WITH EXACTLY ONE Y VALUE.
EVERY X VALUE CAN ONLY HAVE ONE Y VALUE.
LET'S TAKE A LOOK AT A COUPLE OF EXAMPLES
AND SOME WAYS TO REPRESENT RELATIONS.
ONE WAY TO REPRESENT THE RELATIONSHIP BETWEEN TWO SETS
WOULD BE IN THIS MANNER, WHICH IS CALLED MAPPING.
AGAIN, THE MAIN QUESTION IS DOES EVERY X ONLY HAVE ONE Y VALUE?
ON THE LEFT, I HAVE THE DOMAIN.
I THINK OF THIS AS THE SET OF X VALUES.
AND OVER HERE, THESE VALUES WOULD BE THE Y VALUES.
IN FACT, THIS IS DATA THAT REPRESENTS
THE NUMBER OF iPOD SALES ON VARIOUS MONTHS IN 2006
AND 2007.
September 6th IS ONLY PAIRED WITH ONE NUMBER OF SALES.
January 7th IS ONLY PAIRED WITH ONE NUMBER OF SALES AND SO ON.
EVERY X ONLY HAS ONE Y. YES THIS IS A FUNCTION.
OKAY, ON THE NEXT PROBLEM, SAME IDEA.
AGAIN, I THINK OF THE DOMAIN AS THE SET OF X VALUES,
RANGE AS THE SET OF Y VALUES.
AND THE QUESTION--EVERY X ONLY PAIRED WITH ONE Y VALUE.
NOW HERE'S WHERE IT GETS A LITTLE INTERESTING.
4 IS ONLY PAIRED WITH 1 Y VALUE, 16.
AND -4 IS ALSO ONLY PAIRED WITH 1 Y VALUE,
SO IN FACT THIS IS A FUNCTION.
LET'S LOOK AT ONE MORE
THAT'S WRITTEN IN THE MAPPING FORM AS WELL.
THIS IS--OVER HERE ON THE LEFT WE HAVE A DOMAIN,
WHICH WOULD BE THE TOWN.
SO THESE ARE X VALUES.
AND OVER HERE THE RANGE WOULD BE THE Y, WOULD BE THE--
IT LOOKS LIKE BASEBALL TEAMS.
AND THIS DOES FAIL THE FUNCTION TEST BECAUSE CHICAGO,
THIS X VALUE, IS PAIRED WITH TWO DIFFERENT Y VALUES.
SO THIS IS NOT A FUNCTION.
OKAY, NOW ANOTHER WAY TO REPRESENT A RELATION
IS THROUGH A GRAPH.
THE GRAPH OF A FUNCTION F IS A DRAWING THAT REPRESENTS
ALL OF THE INPUT, OUTPUT PAIRS
OR YOU CAN THINK OF AS ORDERED PAIRS.
IN CASES WHERE THE FUNCTION IS GIVEN BY AN EQUATION,
THE GRAPH OF A FUNCTION IS THE GRAPH OF THE EQUATION
Y = F(X).
WHAT THIS IS TRYING TO TELL US IS THIS EQUATION IS
Y = X CUBED - 9X.
SINCE IT IS A FUNCTION, WHAT WE CAN DO IS WE CAN REPLACE THE Y
AS WE NORMALLY WRITE EQUATIONS
WITH THIS SPECIAL NOTATION F(X).
THIS JUST TELLS THE READER THAT THIS GRAPH REPRESENTS A RELATION
THAT IS ALSO A FUNCTION.
EVERY X ONLY HAS ONE Y VALUE.
NOW, WHEN YOU DO HAVE THE GRAPH
AND YOU WANT TO DETERMINE IF IT IS A FUNCTION,
THERE'S A SPECIAL TEST.
THAT TEST IS CALLED THE VERTICAL LINE TEST.
A GRAPH REPRESENTS A FUNCTION
IF IT IS IMPOSSIBLE TO DRAW A VERTICAL LINE
THAT INTERSECTS THE GRAPH THEN MORE THAN ONCE. OKAY.
SO IF I DRAW A VERTICAL LINE WHERE IT'S IMPOSSIBLE--
A GRAPH REPRESENTS A FUNCTION
IF IT IS IMPOSSIBLE TO DRAW A VERTICAL LINE
THAT INTERSECTS THE GRAPH MORE THAN ONCE.
LET'S TAKE A LOOK AT SEVERAL GRAPHS AND TEST THEM.
IS IT IMPOSSIBLE TO DRAW A VERTICAL LINE
THROUGH THIS GRAPH THAT CROSSES IT MORE THAN ONCE?
AND THE ANSWER IS NO.
THERE IS AN EXAMPLE OF WHERE IT CROSSES TWICE.
THIS FAILS THE FUNCTION TEST.
IT'S NOT A FUNCTION, SO WE'LL SAY NO.
NOW, ON THIS GRAPH ON THE RIGHT,
IF I START DRAWING VERTICAL LINES--
NOTICE EVERY VERTICAL LINE ONLY CROSSES THIS GRAPH ONCE.
SO IT PASSES THE FUNCTION TEST. IT IS A FUNCTION.
THE NEXT GRAPH, IF I START DRAWING VERTICAL LINES
I CAN SEE IT IS IMPOSSIBLE TO DRAW A VERTICAL LINE
THAT CROSSES THIS GRAPH MORE THAN ONCE.
THEREFORE, IT PASSES AND IT IS A FUNCTION.
THIS LAST GRAPH, IF I DRAW A VERTICAL LINE HERE,
I JUST CROSSED THE GRAPH IN MORE THAN ONE POINT,
IN FACT TWO POINTS.
THEREFORE IT FAILS AND IT IS NOT A FUNCTION.
I HOPE THAT HELPS EXPLAIN HOW TO DETERMINE
IF A RELATION IS A FUNCTION.
I'LL LEAVE YOU WITH A THOUGHT FOR THE DAY.