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- IN THIS PROBLEM WE'RE GIVEN 3 RELATIONS IN THE FORM OF TABLES,
AS WE SEE HERE BELOW,
AND WE'RE ASKED HERE TO DETERMINE WHICH TABLES
ARE ONE-TO-ONE FUNCTIONS.
TO TEST TO SEE IF A RELATION IS A ONE-TO-ONE FUNCTION,
WE FIRST HAVE TO TEST TO SEE IF IT'S A FUNCTION,
AND THEN TEST TO SEE IF IT'S A ONE-TO-ONE FUNCTION.
SO LET'S REVIEW BOTH DEFINITIONS.
A FUNCTION IS A RELATION
IN WHICH EVERY INPUT HAS EXACTLY ONE OUTPUT.
REMEMBER THE INPUTS ARE THE X VALUES, OR THE DOMAIN,
AND THE OUTPUTS ARE THE Y VALUES, OR THE RANGE.
AN EASIER WAY TO THINK OF THIS IS A RELATION IS A FUNCTION
IF EVERY X HAS EXACTLY 1 CORRESPONDING Y VALUE.
SO ONCE WE KNOW A RELATION IS A FUNCTION,
WE CAN THEN TEST TO SEE IF IT'S A ONE-TO-ONE FUNCTION.
A ONE-TO-ONE FUNCTION IS A FUNCTION
IN WHICH EACH INPUT HAS 1 UNIQUE OUTPUT.
AGAIN, AN EASIER WAY TO THINK OF THIS IS
A FUNCTION IS ONE-TO-ONE
IF EVERY Y HAS EXACTLY 1 CORRESPONDING X.
SO LOOKING AT OUR TABLES,
WE FIRST WANT TO DETERMINE IF THEY ARE FUNCTIONS,
AND THEN IF THEY'RE ONE-TO-ONE FUNCTIONS.
IF THEY'RE NOT FUNCTIONS, THEY CAN'T BE ONE-TO-ONE FUNCTIONS.
AFTER YOU DO THIS A WHILE,
YOU CAN PROBABLY ANSWER THESE QUESTIONS
LEAVING IT IN THE TABLE FORMAT,
BUT WHEN FIRST LEARNING HOW TO DO THIS,
I THINK MAPPING THE TABLE MAKES IT A LOT EASIER.
SO LET'S LOOK AT THIS FIRST TABLE ON THE NEXT SLIDE.
TO MAP THIS RELATION,
WE'RE GOING TO PLACE THIS SET OF X VALUES, OR INPUTS, HERE,
AND THIS SET OF Y VALUES, OR OUTPUTS, HERE,
AND THEN MAP THE RELATION.
SO LOOKING AT THE X VALUES, WE HAVE 3 UNIQUE X VALUES.
WE HAVE -4, -2, AND 1.
EVEN THOUGH 1 OCCURS TWICE, WE ONLY LIST IT ONCE.
THESE ARE THE UNIQUE X VALUES.
AND THEN FOR THE Y VALUES WE HAVE 3, 5, AND 9.
AGAIN, NOTICE HOW 3 OCCURS TWICE.
AND NOW TO MAP THE RELATION, WHEN X = -4, Y = 3,
SO WE'LL DRAW A SEGMENT FROM -4 TO 3.
WHEN X = -2, Y = 5,
SO WE DRAW A SEGMENT FROM -2 TO 5.
WHEN X = 1, Y = 9, SO WE DRAW A SEGMENT HERE.
AND THEN WHEN X = 1, Y IS ALSO 3,
SO WE DRAW ANOTHER SEGMENT FROM 1 TO 3.
NOW THE FUNCTION TEST IS TO MAKE SURE EVERY X
HAS EXACTLY ONE CORRESPONDING Y VALUE.
IN THIS FORM WE CAN EASILY SEE THAT THE X VALUE OF 1
CORRESPONDS TO 2 DIFFERENT Y VALUES.
IT CORRESPONDS TO 3 AND 9.
THEREFORE THIS RELATION IS NOT A FUNCTION,
AND IF IT'S NOT A FUNCTION,
IT CAN'T BE A ONE-TO-ONE FUNCTION EITHER.
SO WE'LL GO AHEAD AND SAY IT'S NOT A FUNCTION,
AND MOVE TO THE NEXT RELATION.
NEXT WE'LL GO AHEAD AND MAP OUR SECOND RELATION.
LOOKING AT THE X VALUES NOTICE HOW THERE'S NO REPETITION.
SO WE HAVE -4, -2, 1, 3.
THERE IS REPETITION IN THE Y VALUES THOUGH.
THE SET OF Y VALUES WILL BE -5, 2, AND 0.
NOW WE'LL MAP OUR RELATION.
-4 MAPS TO -5, -2 MAPS TO 2,
1 MAPS TO 0, AND 3 MAPS TO 2.
WE'LL FIRST TEST TO SEE IF IT'S A FUNCTION,
MEANING EVERY X HAS EXACTLY 1 CORRESPONDING Y,
AND THAT IS TRUE.
NOTICE EVERY X VALUE ONLY HAS 1 SEGMENT MOVING TOWARD THE RIGHT,
SO THIS IS A FUNCTION.
NOW WE'LL SEE IF IT'S A ONE-TO-ONE FUNCTION,
MEANING EVERY Y HAS EXACTLY 1 CORRESPONDING X.
SO LOOKING AT THE Y VALUES NOW,
NOTICE HOW THE Y VALUE OF 2
ACTUALLY HAS 2 CORRESPONDING X VALUES.
THEREFORE, WHILE THIS IS A FUNCTION, IT'S NOT ONE-TO-ONE.
THIS Y VALUE CORRESPONDS TO THE X VALUE OF -2
AND THE X VALUE OF 3.
SO IT'S A FUNCTION, BUT NOT ONE-TO-ONE.
NOW LET'S TEST OUR THIRD RELATION.
NOTICE THERE'S NO REPETITION IN X VALUES OR Y VALUES,
SO THE MAPPING IS VERY STRAIGHTFORWARD.
-2 MAPS TO 3, -1 MAPS TO 4, 0 MAPS TO 5, AND 1 MAPS TO 8.
SO WE'LL FIRST SEE IF IT'S A FUNCTION,
MEANING EVERY X HAS EXACTLY 1 CORRESPONDING Y,
WHICH IT DOES.
NOTICE EVERY X VALUE ONLY HAS 1 SEGMENT MOVING TOWARD THE RIGHT,
SO IT'S A FUNCTION.
AND TO TEST TO SEE IF IT'S A ONE-TO-ONE FUNCTION,
WE WANT TO SEE IF EVERY Y HAS EXACTLY 1 CORRESPONDING X.
LOOKING AT THE Y VALUES,
NOTICE EACH Y VALUE IS ONLY PAIRED WITH 1 X,
BECAUSE THERE'S ONLY 1 SEGMENT LEAVING EACH Y VALUE
MOVING TOWARD THE LEFT.
SO THIS IS A FUNCTION, AND IT'S ALSO A ONE-TO-ONE FUNCTION.
SO YOU MAY HAVE PICKED UP ON THE SHORTCUT HERE.
IF THERE'S NO REPETITION IN THE Y VALUES OR THE X VALUES,
THEN AS A TABLE THE RELATION WOULD BE A ONE-TO-ONE FUNCTION.
BUT HOPEFULLY BY MAPPING THESE,
YOU HAVE A BETTER UNDERSTANDING OF WHAT IT MEANS
TO BE A ONE-TO-ONE FUNCTION.
I HOPE YOU FOUND THIS HELPFUL.