- 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.