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» » Vertical Line Test

Is the Relation a Function?
Using the Vertical Line Test


A function is a relation (a set of ordered pairs) where the value of one variable depends on the value of the other variable. However, in a function, each input (x coordinate) may be paired with only ONE output (y coordinate).

Let's look at a couple of examples to clarify this definition.


Example 1


Let's look at our relation, b that we used in our relations example in the previous lesson.. Is this relation a function? Is each input only paired with only one output?

There are actually two ways to determine if a relation is a function. One way is to analyze the ordered pairs, and the other way is to use the vertical line test. Let's analyze our ordered pairs first.


identifying functions

Since each input has a different output, this can be classified as a function.

Let's verify it with the vertical line test. This special test is used when you graph the ordered pairs. You imagine a vertical line being drawn through the graph. If the vertical line only touches the graph at one point, then it is a function. If the vertical line touches in more than one point, then it is NOT a function.

Let's graph our points and use the vertical line test to prove that this is a function. I drew the vertical lines (output) on the graph to demonstrate what it would look like. Unless asked, you really don't need to draw the vertical lines, you can just imagine the vertical line, or I have my students use the edge of a sheet of paper and move it across the graph.

vertical line test example

Since each vertical line only touches the graph at one point, this relation can be classified as a function.




Let's take a look at another example.


Example 2


Is the relation s, a function?


s:{(-3, 2) (-1, 6) (1,2)

Let's ask ourselves: Is each input paired with only one output? (This one is a little tricky!)

YES, each input is paired with only one output. I know that you see an output of 2 twice, but each is paired with a different input. The outputs can be the same, as long as the inputs are different. Let's look at the vertical line test.

vertical line test

Yes, this is a function - remember that the outputs can be duplicated as long as the inputs are different.




Ok... I hope that you aren't confused by that last example. If you are, take a look at this last example. You will see a relation that is NOT considered a function. Hopefully this will help build your understanding.



Example 3


Is the relation, c, a function?

Let's ask ourselves: Is each input paired with only one output?

c:{(3,3) (-1,0) (3,-3)

NO, look at the ordered pairs - there are 2 ordered pairs with input of 3 and they have different ranges! Think about our little function machine - you cannot put in a 3 and get out a 0, and then put in another 3 and get out a -3. Each time you put in a 3, you should get the same answer.

Therefore, this relation is NOT a function. Let's look at the vertical line test.

identifying funcitons with vertical line test



So, how do you feel about identifying whether or not a relation is a function? I know it can be a little confusing.

Just remember: If you have duplicate inputs and they are paired with different outputs, then the relation is not a function. The vertical line test is always a good technique to use if you are unsure or want to justify your answer.

If you need more direct instruction, check out the Algebra Class E-course for video tutorials and more practice problems.



vertical line test practice problem
Click here to practice identifying functions using this special test.




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