- WELCOME TO A LESSON ON THE VERTICAL LINE TEST.
A FUNCTION IS A SPECIAL RELATION IN WHICH EACH INPUT,
NORMALLY X, IS EXACTLY ONE OUTPUT,
NORMALLY Y.
SO WHEN GIVEN THE GRAPH OF A RELATION,
THE VERTICAL LINE TEST CAN QUICKLY DETERMINE
IF THE RELATION IS ALSO A FUNCTION.
THE VERTICAL LINE TEST, STATES THAT IF A VERTICAL LINE
NEVER INTERSECTS THE GRAPH OF A RELATION IN MORE
THAN ONE POINT THEN THE RELATION
IS ALSO A FUNCTION.
SO WHAT WE'LL DO IS SKETCH THE VERTICAL LINES
ACROSS EACH GRAPH TO DETERMINE IF IT PASSES
THE VERTICAL LINE TEST TO SEE IF IT'S A FUNCTION
OR NOT.
SO LOOKING AT OUR FIRST GRAPH, NOTICE HOW IF WE SKETCH
SEVERAL VERTICAL LINES, NONE OF THEM WOULD INTERSECT
THE GRAPH IN MORE THAN ONE POINT
VERIFYING THAT FOR EVERY INPUT IS EXACTLY ONE OUTPUT.
SO THE ANSWER IS, YES, THIS IS A FUNCTION.
NOW, IF WE COMPARE THIS TO THE SECOND GRAPH,
NOTICE AS WE SKETCH THE VERTICAL LINES,
THESE VERTICAL LINES INTERSECT THE GRAPH
IN MORE THAN ONE POINT, SPECIFICALLY TWO POINTS.
AND THEREFORE, THIS FAILS THE VERTICAL LINE TEST
AND IT'S NOT A FUNCTION, SO THE ANSWER IS, NO,
IT'S NOT A FUNCTION.
AND MORE SPECIFICALLY, IF WE CONSIDER THE X VALUE OF,
LET'S SAY, 5, NOTICE HOW WHEN X IS 5
THERE ARE TWO CORRESPONDING Y VALUES,
ONE HERE ABOVE THE X AXIS AND ONE HERE BELOW THE X AXIS.
SINCE THIS INPUT HAS TWO OUTPUTS,
THIS IS WHY IT FAILS THE VERTICAL LINE TEST
AND IS NOT A FUNCTION.
LOOKING AT OUR NEXT GRAPH, AS WE SKETCH VERTICAL LINES
EVERYTHING LOOKS GOOD UNTIL WE GET TO THIS POINT HERE
WHERE IT MAY BE HARD TO TELL, BUT THERE'S A HOLE HERE,
AND THERE'S A POINT DOWN HERE.
SO WHEN WE SKETCH A VERTICAL LINE
AT X = 2 HERE IT STILL ONLY HAS ONE POINT OF INTERSECTION
BECAUSE THERE'S A HOLE HERE AND THERE'S A CLOSED POINT HERE.
SO WHEN X = 2 THE FUNCTION VALUE IS ACTUALLY DOWN HERE AT Y = -2.
THEREFORE, THESE VERTICAL LINES NEVER INTERSECT THE GRAPH
IN MORE THAN ONE POINT, AND THEREFORE THIS PASSES
THE VERTICAL LINE TEST AND THE ANSWER IS, YES,
IT'S A FUNCTION.
WE COULD ALSO APPLY THE VERTICAL LINE TEST
WHEN WE HAVE POINTS GRAPHED ON THE COORDINATE PLANE
AS WE SEE HERE IN THE NEXT TWO EXAMPLES.
NOTICE IF SKETCH VERTICAL LINES
ANYWHERE THROUGHOUT THIS COORDINATE PLANE,
IT NEVER INTERSECT THE POINTS IN MORE THAN ONE POINT,
AND THEREFORE IT PASSES THE VERTICAL LINE TEST
AND THE ANSWER IS, YES, THIS IS A FUNCTION.
EVERY INPUT IS EXACTLY ONE OUTPUT.
NOW, LOOKING AT THE NEXT GRAPH THOUGH, IT LOOKS GOOD
ALL THE WAY THROUGH HERE, BUT THEN NOTICE THAT X = 4 HERE,
THIS VERTICAL LINE WOULD INTERSECT THE GRAPH
IN TWO POINTS, ONE HERE AT Y = 0 AND ONE HERE AT Y = 5.
THEREFORE, THERE ARE TWO POINTS OF INTERSECTION
AND THIS IS NOT A FUNCTION.
STILL A RELATION, BUT IT'S NOT A FUNCTION.
THE X VALUE OF 4 HAS ACTUALLY TWO OUTPUTS.
AND THE NEXT GRAPH HERE, WE CAN EASILY SEE
THESE VERTICAL LINES INTERSECT THE GRAPH
IN MORE THAN ONE POINT, THEREFORE THIS FAILS
AND IS NOT A FUNCTION, SO OUR ANSWER IS NO.
LET'S TAKE A LOOK AT A FEW MORE.
SKETCH SOME VERTICAL LINES HERE.
NONE OF THESE LINES INTERSECT THE GRAPH
IN MORE THAN ONE POINT.
NOTICE HOW THERE IS A BREAK IN THE GRAPH, SO WHEN X = 0
THERE ARE 0 POINTS OF INTERSECTION WITH THE GRAPH.
THAT'S FINE BECAUSE 0 IS NOT MORE THAN 1,
THEREFORE THIS PASSES A VERTICAL LINE TEST
AND, YES, IT'S A FUNCTION.
NEXT YOU CAN EASILY SEE THIS GRAPH HERE PASSES
THE VERTICAL LINE TEST.
WE NEVER HAVE MORE THAN ONE POINT OF INTERSECTION,
AND THEREFORE THE ANSWER IS, YES, IT'S A FUNCTION.
AND THEN FINALLY ON THIS GRAPH HERE,
AGAIN THERE ARE VERTICAL ASYMPTOTES, BUT AT NO POINT
DOES A VERTICAL LINE INTERSECT THE GRAPH
IN MORE THAN ONE POINT.
AND THEREFORE, IT PASSES AND THIS RELATION IS A FUNCTION,
SO THE ANSWER IS YES.
NOW, IT MAY BE HARD TO SEE, BUT IN THIS GRAPH
THIS IS AN OPEN POINT, SO AS WE SKETCH VERTICAL LINES
ACROSS THIS GRAPH NOTICE X = -1 HERE.
BECAUSE WE HAVE A HOLE HERE AND A CLOSED POINT HERE
THERE'S ONLY ONE POINT OF INTERSECTION.
THEREFORE, THIS PASSES THE VERTICAL LINE TEST,
AND ONCE AGAIN, THE ANSWER IS YES, THIS IS A FUNCTION.
LOOKING AT THE NEXT GRAPH, EVERYTHING LOOKS GOOD UNTIL,
AGAIN, WE REACH X = -1.
NOTICE HOW THERE ARE TWO CLOSED POINTS HERE AND HERE,
SO WHEN X = -1 THERE ARE ACTUALLY
TWO POINTS OF INTERSECTION, WHICH IS MORE THAN ONE,
AND THEREFORE THIS FAILS THE VERTICAL LINE TEST,
AND THE ANSWER IS, NO, IT'S NOT A FUNCTION.
WHEN X = -1 THERE ARE ACTUALLY
TWO OUTPUTS, NOT JUST ONE.
LOOKING AT OUR LAST EXAMPLE, AS WE SKETCH VERTICAL LINES,
IT LOOKS GOOD UNTIL, AGAIN, WE REACH X = -1 HERE.
NOTICE HOW THERE'S NO HOLE IN THE GRAPH DOWN HERE.
THERE'S ALSO A POINT UP HERE AT Y = 2.
SO THIS VERTICAL LINE INTERSECTS THE GRAPH
IN MORE THAN ONE POINT, ACTUALLY TWO POINTS,
SO THIS FAILS THE VERTICAL LINE TEST,
AND THE ANSWER IS, NO, THIS IS NOT A FUNCTION.
I HOPE YOU FOUND THIS HELPFUL.