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- WELCOME BACK FOR SOME MORE EXAMPLES OF DETERMINING
IF A GRAPH REPRESENTS A ONE-TO-ONE FUNCTION.
HOPEFULLY YOU ALREADY WATCHED PART 1.
IF NOT, THERE'S A LINK ON THE SCREEN.
TO TEST TO SEE IF A GRAPH IS A ONE-TO-ONE FUNCTION,
WE FIRST WANT TO TEST TO SEE IF THE GRAPH REPRESENTS A FUNCTION
BY USING THE VERTICAL LINE TEST.
WE CROSS A VERTICAL LINE ACROSS THE GRAPH --
DOES NOT INTERSECT THE GRAPH IN MORE THAN ONE POINT
IT PASSES THE VERTICAL LINE TEST AND IS A FUNCTION.
ONCE WE KNOW THE GRAPH REPRESENTS A FUNCTION,
WE CAN USE THE HORIZONTAL LINE TEST
TO SEE IF THE FUNCTION IS A ONE-TO-ONE FUNCTION.
IT'S THE SAME TEST
BUT WE USE A HORIZONTAL LINE INSTEAD OF A VERTICAL LINE.
IF A HORIZONTAL LINE DOES NOT INTERSECT THE GRAPH
IN MORE THAN ONE POINT,
IT PASSES AND THE FUNCTION IS A ONE-TO-ONE FUNCTION.
LET'S GO THROUGH SEVERAL MORE EXAMPLES.
WE FIRST WANT TO CHECK TO SEE IF THE GIVEN GRAPH HERE
IS A FUNCTION.
NOTICE IF WE SKETCH SEVERAL VERTICAL LINES ACROSS THIS GRAPH
THEY WOULD NOT INTERSECT THE GRAPH IN MORE THAN ONE POINT,
WHICH MEANS IT PASSES THE VERTICAL LINE TEST
AND THIS GRAPH IS A FUNCTION.
SO NOW THAT WE KNOW IT'S A FUNCTION,
WE CAN TEST TO SEE IF IT'S A ONE-TO-ONE FUNCTION.
LET'S JUST MAKE A NOTE HERE THAT IT IS A FUNCTION.
NOW I'LL PERFORM THE SAME TEST BUT WE'LL USE A HORIZONTAL LINE.
SO WE'LL SKETCH SEVERAL HORIZONTAL LINES
ACROSS THIS GRAPH.
NOTICE HOW THESE LINES DO INTERSECT THE RED GRAPH
IN MORE THAN ONE POINT.
THEREFORE IT FAILS THE HORIZONTAL LINE TEST
AND THIS IS NOT A ONE-TO-ONE FUNCTION.
SO IT IS A FUNCTION, BUT IT'S NOT ONE-TO-ONE.
LET'S GO AHEAD AND TEST THIS GRAPH HERE NOW.
AGAIN WE'RE GOING TO START WITH THE VERTICAL LINE TEST.
SO WE'LL SKETCH SEVERAL VERTICAL LINES ACROSS THIS GRAPH.
NOTICE RIGHT HERE IN THE MIDDLE OF THESE VERTICAL LINES
ARE INTERSECTING THE GRAPH IN MORE THAN ONE POINT.
THEREFORE, IT FAILS THE VERTICAL LINE TEST
WHICH MEANS THIS IS NOT A FUNCTION.
IF IT'S NOT A FUNCTION,
IT'S NOT GOING TO BE A ONE-TO-ONE FUNCTION EITHER.
SO FAR NEITHER OF THESE TWO ARE ONE-TO-ONE FUNCTIONS.
NOW LET'S LOOK AT THIS GRAPH HERE,
IT'S A PIECE WISE DEFINED FUNCTION,
THE RIGHT SIDE OF EACH OF THESE SEGMENTS IS AN OPEN CIRCLE.
SO WE'LL START WITH THE VERTICAL LINE TEST.
NOW IT LOOKS LIKE AT THE END OF THESE SEGMENTS,
IT WOULD FAIL THE VERTICAL LINE TEST
BUT IT'S NOT GOING TO BECAUSE AS I MENTIONED A MOMENT AGO,
NOW LET'S SAY X = 5,
WE HAVE AN OPEN POINT HERE AND A CLOSED POINT HERE,
SO THE FUNCTION IS NOT DEFINED ON THIS OPEN POINT.
IT'S DEFINED ON THE CLOSED POINT
SO IT DOES PASS THE VERTICAL LINE TEST AND IS A FUNCTION.
SO WE CAN GO AHEAD AND TEST
TO SEE IF IT'S ALSO A ONE-TO-ONE FUNCTION.
SO NOW WE'LL PERFORM THE HORIZONTAL LINE TEST.
NOTICE WHEN THE HORIZONTAL LINES INTERSECT THOSE SEGMENTS,
THERE WOULD BE AN INFINITE NUMBER OF POINTS ON THAT SEGMENT
AND THEREFORE THESE HORIZONTAL LINES DO INTERSECT THE GRAPH
IN MORE THAN ONE POINT
SO AGAIN WHILE THIS IS A FUNCTION,
IT'S NOT A ONE-TO-ONE FUNCTION.
SO WE ONLY HAVE ONE MORE GRAPH TO TEST.
SO WE'LL START WITH THE VERTICAL LINE TEST.
NOTICE NONE OF THESE VERTICAL LINES INTERSECT THE GRAPH
IN MORE THAN ONE POINT
THEREFORE, IT PASSES THE VERTICAL LINE TEST
AND THE GRAPH IS A FUNCTION.
AND NOW TO TEST TO SEE IF IT'S A ONE-TO-ONE FUNCTION,
WE'LL USE THE HORIZONTAL LINE TEST.
SO WE'LL SKETCH SEVERAL HORIZONTAL LINES
ACROSS THIS GRAPH.
NOTICE NONE OF THESE WOULD INTERSECT THE GRAPH
IN MORE THAN ONE POINT
THEREFORE THIS FUNCTION
PASSES THE HORIZONTAL LINE TEST AS WELL,
MEANING THIS GRAPH IS ALSO A ONE-TO-ONE FUNCTION.
SO OF ALL THESE GRAPHS, THIS IS THE ONLY ONE-TO-ONE FUNCTION.
I HOPE THESE EXAMPLES WERE HELPFUL.