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- WE WANT TO DETERMINE WHICH OF THE FOLLOWING TABLES
REPRESENT FUNCTIONS.
A FUNCTION IS RELATION IN WHICH EVERY INPUT
HAS EXACTLY ONE OUTPUT.
SO LOOKING AT THESE TABLES, THE X VALUES WILL BE THE INPUTS
AND THE Y VALUES WILL BE THE OUTPUTS.
SO IF THESE TABLES ARE FUNCTIONS,
WE COULD SAY THAT EVERY X VALUE
MUST HAVE EXACTLY ONE CORRESPONDING Y VALUE.
IF THIS IS NOT TRUE, THE TABLE WOULD NOT
BE A FUNCTION.
SO WE WANT TO ANALYZE THESE TABLES AND MAKE SURE
EVERY X VALUE HAS EXACTLY ONE CORRESPONDING Y VALUE.
SO WE TAKE A LOOK AT THIS FIRST TABLE,
NOTICE THAT THE INPUT OF 2 OCCURS TWICE
WITH TWO DIFFERENT OUTPUTS.
HERE WE HAVE AN INPUT OF 2 WITH AN OUTPUT OF 3,
AND HERE WE HAVE AN INPUT OF 2 WITH AN OUTPUT OF -1.
THIS DOES NOT SATISFY THE DEFINITION OF A FUNCTION
BECAUSE THE X VALUE OF 2 ACTUALLY HAS TWO DIFFERENT
OUTPUTS, OR TWO CORRESPONDING Y VALUES.
THEREFORE, THIS WOULD NOT BE A FUNCTION.
TO HELP VISUALIZE THIS IT'S OFTEN HELPFUL TO SHOW
THE RELATIONSHIP BETWEEN X AND Y
USING MAPPING, WHICH MEANS YOU PUT THE SET OF INPUTS HERE
OR THE DOMAIN HERE, AND THE SET OF OUTPUTS
OR THE RANGE HERE, AND THEN DRAW ARROWS
TO SHOW EACH INPUT WITH THE CORRESPONDING OUTPUT.
SO NOTICE OUR INPUTS WOULD BE 2, -1 -4, AND 5.
WE WOULD NOT LIST 2 TWICE, SO WE HAVE 2, -1, -4, AND 5.
AND THE OUTPUTS ARE THE Y VALUES,
SO WE HAVE 3, 4, 0, -1, AND 7.
SO WHEN THE INPUT IS 2, HERE THE OUTPUT IS 3.
SO WE SHOW THAT RELATIONSHIP BY DRAWING AN ARROW FROM 2 TO 3.
WHEN THE INPUT IS -1 THE OUTPUT IS 4.
AND WE'LL OFTEN SAY -1 MAPS TO 4,
SO WE CAN SAY -4 MAPS TO 0.
BUT NOTICE HOW 2 ALSO MAPS TO -1,
SO WE'D DRAW ANOTHER ARROW FROM 2 TO -1,
AND THEN 5 MAPS TO 7.
SO NOTICE THAT 2 HAS TWO OUTPUTS,
ONE HERE AND ONE HERE.
AND THIS IS THE REASON WHY THIS IS NOT A FUNCTION.
NOW LETS TAKE A LOOK AT THE SECOND TABLE.
NOTICE THAT NONE OF THE INPUTS HERE REPEAT THEMSELVES,
THEREFORE THERE'S NO WAY EVERY X CAN HAVE MORE THAN ONE
CORRESPONDING Y VALUE.
SO THIS WILL BE A FUNCTION, BUT WE'LL ALSO MAP IT
TO ILLUSTRATE IT BETTER.
SO, AGAIN, WE'LL PUT THE INPUTS OR THE ELEMENTS OR NUMBERS
IN THE DOMAIN HERE, AND THEN WE'LL PUT
THE ELEMENTS OR THE NUMBERS AND THE RANGE HERE.
SO THE INPUTS WOULD BE 4, -3, 1, 2, 5, AND NOTICE THERE IS SOME
REPETITION IN THE OUTPUTS.
THERE'S ONLY THREE OUTPUTS, -5, -2, AND 0.
NOW LET'S GO AHEAD AND SHOW THE MAPPING.
IF AN INPUT OF 4 THE OUTPUT IS -5,
SO 4 MAPS TO -5, -3 MAPS TO -2,
1 ALSO MAPS TO -2, 2 ALSO MAPS TO -2,
AND 5 MAPS TO 0.
NOW, THIS DOES LOOK STRANGE BECAUSE WE HAVE THREE ARROWS
ALL POINTING TO -2 IN THE RANGE, BUT NOTICE HOW EVERY INPUT
OR EVERY VALUE HERE DOES HAVE EXACTLY ONE OUTPUT,
EVEN THOUGH AT TIMES THEY DO HAVE THE SAME OUTPUT.
SO BECAUSE EVERY X HAS EXACTLY ONE Y
THIS IS A FUNCTION.
NOW LET'S TAKE A LOOK AT ONE MORE EXAMPLE HERE.
THE FIRST THING WE SHOULD NOTICE IN THIS TABLE IS THAT THE INPUT
OF -1 DOES OCCUR TWICE.
SO -1 DOES HAVE TWO OUTPUTS OR CORRESPONDING Y VALUES.
THEREFORE, AGAIN, THIS TABLE IS NOT A FUNCTION.
AND LET'S GO AHEAD AND MAP THIS AS WELL.
SO THE INPUTS ARE 0, -1, 4, AND 2.
AGAIN, WE DO NOT LIST -1 TWICE.
AND THE OUTPUTS WILL BE 3, 4, 0, -1, 1.
SO 0 MAPS TO 3, -1 MAPS TO 4,
4 MAPS TO 0, 2 MAPS TO -1,
AND THEN -1 MAPS TO 1.
SO FROM -1 WE HAVE ANOTHER ARROW GOING
FROM -1 TO +1.
SO, AGAIN, BECAUSE THIS INPUT OF -1 HAS ACTUALLY TWO OUTPUTS,
THIS TABLE DOES NOT REPRESENT A FUNCTION.
I HOPE YOU FOUND THESE EXAMPLES HELPFUL.