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- IN THIS EXERCISE, WE WANT TO KNOW WHICH GRAPHS REPRESENT
ONE-TO-ONE FUNCTIONS.
TO TEST TO SEE IF A GRAPH IS A ONE-TO-ONE FUNCTION
WE FIRST HAVE TO SEE IF THE GRAPH IS A FUNCTION
AND THEN TEST TO SEE IF IT'S A ONE-TO-ONE FUNCTION.
GRAPHICALLY, THE VERTICAL LINE TEST CAN BE USED TO SEE
IF A GRAPH REPRESENTS A FUNCTION.
TO PERFORM THE VERTICAL LINE TEST
WE PASS A VERTICAL LINE ACROSS THE GRAPH
AND IF IT DOES NOT INTERSECT THE GRAPH IN MORE THAN ONE POINT,
IT PASSES THE VERTICAL LINE TEST AND IS A FUNCTION.
ONCE WE KNOW A GRAPH IS A FUNCTION,
WE CAN TEST TO SEE IF IT'S A ONE-TO-ONE FUNCTION
USING THE HORIZONTAL LINE TEST.
A SIMILAR TEST BUT WE USE A HORIZONTAL LINE INSTEAD.
TO SEE IF THE GRAPH OF A FUNCTION IS ONE-TO-ONE
WE PASS A HORIZONTAL LINE ACROSS THE GRAPH
AND IF IT DOES NOT INTERSECT THE GRAPH IN MORE THAN ONE POINT
THEN IT PASSES THE HORIZONTAL LINE TEST
AND IS A ONE-TO-ONE FUNCTION.
SO WE ALWAYS WANT TO PERFORM THE VERTICAL LINE TEST FIRST
AND THEN THE HORIZONTAL LINE TEST.
IF THE GRAPH DOES NOT REPRESENT A FUNCTION,
IT CANNOT BE A ONE-TO-ONE FUNCTION.
SO LOOKING AT OUR FIRST EXAMPLE HERE
LET'S PERFORM THE VERTICAL LINE TEST.
SO I'LL SKETCH SEVERAL VERTICAL LINES ACROSS THE GRAPH.
WE CAN SEE THESE LINES WOULD NEVER INTERSECT THE GRAPH
IN MORE THAN ONE POINT,
WHICH MEANS THIS BLUE GRAPH IS A FUNCTION.
SO IT IS A FUNCTION
SO NOW WE CAN TEST TO SEE IF IT'S A ONE-TO-ONE FUNCTION
BY PERFORMING THE HORIZONTAL LINE TEST.
NOW WE'LL SKETCH SEVERAL HORIZONTAL LINES
ACROSS THIS GRAPH.
WE CAN QUICKLY SEE THESE HORIZONTAL LINES
DO INTERSECT THE GRAPH IN MORE THAN ONE POINT
THEREFORE IT FAILS THE HORIZONTAL LINE TEST
AND IS NOT A ONE-TO-ONE FUNCTION.
SO THIS BLUE GRAPH IS A FUNCTION BUT IT'S NOT A ONE-TO-ONE.
LET'S LOOK AT THE GRAPH HERE NOW.
AGAIN, WE'LL START WITH THE VERTICAL LINE TEST.
NOTICE THESE VERTICAL LINES DO INTERSECT THE GRAPH
IN MORE THAN ONE POINT
WHICH MEANS THIS GRAPH IS NOT A FUNCTION
AND IF IT'S NOT A FUNCTION
IT OBVIOUSLY CAN'T BE A ONE-TO-ONE FUNCTION.
SO THESE FIRST TWO GRAPHS ARE OUT,
NEITHER OF THEM ARE ONE-TO-ONE FUNCTIONS.
LET'S TAKE A LOOK AT THIS GRAPH HERE.
AGAIN, WE'LL START WITH THE VERTICAL LINE TEST.
NOTICE THESE VERTICAL LINES DO NOT INTERSECT THE GRAPH
IN MORE THAN ONE POINT,
THEREFORE IT PASSES THE FUNCTION TEST
SO WE CAN SAY IT'S A FUNCTION
AND NOW WE'LL PERFORM THE HORIZONTAL LINE TEST
TO SEE IF IT'S A ONE-TO-ONE FUNCTION.
NOTICE AS WE PASS HORIZONTAL LINES ACROSS THIS GRAPH
THEY DO NOT INTERSECT THE GRAPH IN MORE THAN ONE POINT.
LOOKS LIKE IT MIGHT BE CLOSE NEAR THE ORIGIN
BUT IT DOES NOT INTERSECT THE GRAPH IN MORE THAN ONE POINT
THEREFORE IT PASSES THE HORIZONTAL LINE TEST
AND THIS IS NOT ONLY A FUNCTION
THIS IS ALSO A ONE-TO-ONE FUNCTION.
AND THEN FOR OUR LAST GRAPH
WE'LL START WITH A VERTICAL LINE TEST
AND AGAIN WE CAN QUICKLY SEE
THIS FAILS THE VERTICAL LINE TEST
BECAUSE THESE VERTICAL LINES INTERSECT THE GRAPH
IN TWO POINTS
THEREFORE IT'S NOT A FUNCTION.
THEREFORE, IT CANNOT BE ONE-TO-ONE.
SO OF THESE FOUR GRAPHS
THE ONLY GRAPH THAT REPRESENTS A ONE-TO-ONE FUNCTION
IS THIS GRAPH HERE.
OKAY, WE'LL LOOK AT SOME MORE EXAMPLES IN THE NEXT VIDEO.