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- WE WANT TO DETERMINE IF THE GIVEN GRAPHS
REPRESENT FUNCTIONS.
TO DO THIS WE CAN USE SOMETHING CALLED A VERTICAL LINE TEST.
BEFORE WE EXPLAIN THIS, LET'S REVIEW THE DEFINITION
OF A FUNCTION.
A FUNCTION IS A RELATION IN WHICH EVERY INPUT OR VALUE
IN THE DOMAIN IS PAIRED WITH EXACTLY ONE OUTPUT
OR VALUE IN THE RANGE.
TO DETERMINE IF A GRAPH SATISFIES THIS CONDITION,
WE CAN USE SOMETHING CALLED A VERTICAL LINE TEST.
VERTICAL LINE TEST SAYS IF A VERTICAL LINE
INTERSECTS A GRAPH IN MORE THAN ONE POINT,
THE GRAPH FAILS THE VERTICAL LINE TEST
AND, THEREFORE, IS NOT A FUNCTION.
SO LOOKING AT THIS FIRST GRAPH, IF WE SKETCH SEVERAL
VERTICAL LINES WE CAN SEE NO VERTICAL LINE WOULD EVER
INTERSECT THIS GRAPH IN MORE THAN ONE POINT.
AND, THEREFORE, IT PASSES THE VERTICAL LINE TEST
AND THIS GRAPH DOES REPRESENT A FUNCTION.
BUT IF WE TAKE A LOOK AT THE GRAPH BELOW,
IF WE SKETCH SEVERAL VERTICAL LINES,
AT FIRST THESE VERTICAL LINES ONLY INTERSECT THE GRAPH
IN ONE POINT.
BUT THEN HERE THEY INTERSECT THE GRAPH
IN ACTUALLY THREE POINTS, THEREFORE THIS FAILS
THE VERTICAL LINE TEST AND IS NOT A FUNCTION.
TO SEE WHY, IF WE CONSIDER THE INPUT OF LET'S SAY X = 0,
NOTICE HOW WHEN X = 0 THERE ARE ONE, TWO, THREE
CORRESPONDING Y VALUES, THEREFORE THIS DOES NOT SATISFY
THE DEFINITION OF A FUNCTION AND IS NOT A FUNCTION.
NOW, WE NEED TO BE CAREFUL WITH THIS NEXT GRAPH.
NOTICE HOW IT CONSISTS OF SEVERAL SEGMENTS,
BUT THE RIGHT END POINTS OF THE SEGMENTS
ARE OPEN CIRCLES.
SO TO EMPHASIZE THIS I'M GOING TO MAKE
LARGER OPEN POINTS ON THE RIGHT OF EACH SEGMENT.
SO FOR EXAMPLE, IF WE WANTED TO DETERMINE
THE FUNCTION VALUE AT 2, THE FUNCTION VALUE
WOULD BE UP HERE AT +2 NOT DOWN HERE AT +1.
AND THAT'S IMPORTANT BECAUSE WHEN WE APPLY
THE VERTICAL LINE TEST WE CAN SEE IN THE MIDDLE
OF THESE SEGMENTS THERE'S ONLY GOING
TO BE ONE POINT OF INTERSECTION.
BUT AT INTEGER VALUES OF X WE NEED TO BE A LITTLE BIT
MORE CAREFUL.
FOR EXAMPLE, RIGHT HERE AT X = 3 THE VERTICAL LINE
MAY APPEAR TO INTERSECT THE GRAPH IN MORE
THAN ONE POINT, BUT BECAUSE THIS IS AN OPEN POINT
AND THIS IS A CLOSED POINT, THIS VERTICAL LINE
ONLY INTERSECTS THE GRAPH AT THIS POINT HERE
NOT RIGHT HERE.
THEREFORE, THE VERTICAL LINES WILL ONLY INTERSECT THIS GRAPH
AT ONE POINT, AND THEREFORE THIS DOES PASS
THE VERTICAL LINE TEST AND IS A FUNCTION.
NOW, I DO WANT TO MENTION IF THE GRAPH LOOKS SOMETHING
LIKE THIS WHERE BOTH END POINTS WERE CLOSED, THIS WOULD FAIL
THE VERTICAL LINE TEST BECAUSE IF THIS WAS THE GRAPH
AND WE SKETCHED A VERTICAL LINE HERE,
IT DOES INTERSECT THE GRAPH AT THIS POINT AND THIS POINT.
AND THEREFORE, THIS X VALUE WOULD HAVE TWO CORRESPONDING
Y VALUES AND WOULD NOT BE A FUNCTION.
BUT NOTICE THAT OUR FUNCTION HAD OPEN POINTS
ON THE RIGHT SIDE OF EACH SEGMENT.
AND THEN FOR OUR LAST EXAMPLE, THIS IS MUCH MORE
STRAIGHT FORWARD.
EVERY VERTICAL LINE WOULD ONLY INTERSECT
THIS GRAPH AT ONE POINT, AND THEREFORE
THIS IS A FUNCTION.
OKAY. I HOPE YOU FOUND THIS HELPFUL.
THIS ONE HERE WAS A LITTLE BIT MORE INVOLVED,
BUT WE WERE STILL ABLE TO APPLY THE VERTICAL LINE TEST.