What is a Function? - Applying the Vertical Line Test

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  • 0:10 What Is a Function?
  • 2:51 The Vertical Line Test
  • 3:43 Applying the Vertical…
  • 5:04 Lesson Summary
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Lesson Transcript
Instructor: Tyler Cantway

Tyler has tutored math at two universities and has a master's degree in engineering.

Equations come in all shapes and sizes. Trying to determine which equations are functions can be difficult. You can apply the vertical line test to a graph to quickly tell whether or not it is a function.

What is a Function?

If you were going to give someone directions to your favorite restaurant, how would you do it? Some people might give directions using street names. Some people might do the same but use landmarks. These can be very descriptive, but oftentimes you have to already know a lot of other information to make these directions useful. If I gave you directions to my favorite pizza shop, it would go something like this: it's downtown, go down 3rd Street, turn right on Market and it's on your left near the big bridge. These directions are good, but only if you know what city I live in and how to get to 3rd Street in the first place. In other words, you really have to understand a lot of other information to make this useful.

To make things easier, many people prefer to use a map. A map gives you information like where you are, where the pizza shop is, distance, direction, city location and the location of a nearby sandwich place just in case you don't feel like eating pizza. A map shows you how the roads intersect, where landmarks lie and other things that might not be easy to notice when you are just looking at a list of streets.

Let's try to understand the difference in more mathematical terms. In math, when we write out numbers and variables with an equals sign, we call it an equation. This is a very general term, and all it really means is the left side equals the right side. We use the term function to describe a special relationship between two variables where each input value is paired with exactly one output value. What this means is, if you choose any x value and evaluate it, you will get a single specific y value. For example, the equation y=x+2 has an x variable and a y variable. If we input x=1, we get y=1+2, which simplifies to y=3. No matter what, this never changes. If x=1, y=3. This is what we call a function.

Equations can be like written directions. They contain the bare minimum information. They can come in any shape, size or complexity. It's often hard to tell by looking, so you have to look for clues to understand what they are really saying. You really have to be good to know which equations are functions. But, just like going from written directions to a map, there is a quick way to see all the information at once and tell which ones are functions.

The Vertical Line Test

If we use a map, we can see almost everything. This is the same for using a graph. If we took the graph of an equation, there is an easy way to tell if you are looking at a function. We call this the vertical line test. The vertical line test works like this: look at the graph and see if there is any vertical line that touches more than one place on the graph. To do this simply, when you are looking at the graph of an equation, lay your pencil vertically and slide it to the left and right. See if the graph touches the pencil in more than one place. This is a simple test that works on even the most complicated graphs.

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