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Vertical Line Test

Not a function if a vertical line crosses graph in 2+ places

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Relations and Functions

The following table of values represents data collected by a student in a math class.

Does this set of ordered pairs represent a function?

Guidance

Consider the relationship between two variables. You can think of this relationship in terms of an input/output machine.

If there is only one output for every input, you have a function. If not, you have a relation. Relations can have more than one output for every input. A relation is any set of ordered pairs. A function is a set of ordered pairs where there is only one value of for every value of .

Look at the two tables below. Table A shows a relation that is a function because every value has only one value. Table B shows a relation that is not a function because there are two different values for the value of 0.

Table A
0 4
1 7
2 7
3 6
Table B
0 4
0 2
2 6
2 7

When looking at the graph of a relation, you can determine whether or not it is a function using the vertical line test. If a vertical line can be drawn anywhere through the graph such that it intersects the graph more than once, the graph is not function.

Example A

Determine if the following relation is a function.

4 3.6
7.8 7.2

Solution:

The relation is a function because there is only one value of for every value of .

Example B

Which of the following graphs represent a function?

Solution:

In order to answer this question, you need to use the vertical line test. A graph represents a function if no vertical line intersects the graph more than once. Let’s look at the first graph. Draw a vertical line through the graph.

Since the vertical line hit the graph more than once (indicated by the two red dots), the graph does not represent a function.

Since the vertical line hit the graph only once (indicated by the one red dot), the graph does represent a function.

Since the vertical line hit the graph only once (indicated by the one red dot), the graph does represent a function.

Since the vertical line hit the graph more than once (indicated by the three red dots), the graph does not represent a function.

Example C

Which of the following represent functions?

Solution:

a) This is a function because every input has only one output.

b) This is not a function because one input (1) has two outputs (2 and 7).

c) This is a function because every input has only one output.

Concept Problem Revisited

If you look at this table, there are two places where you see the more than one output for a single input.

You can conclude that this set of ordered pairs does not represent a function. It is just a relation.

Vocabulary

Function
A function is an example of a relation where there is only one output for every input. In other words, for every value of , there is only one value for .
Relation
A relation is any set of ordered pairs . A relation can have more than one output for an input.
Vertical Line Test
The Vertical Line Test is a test for functions. If you can take your pencil and draw a straight vertical line through any part of the graph, and the pencil hits the graph more than once, the graph is not a function.

Guided Practice

1. Is the following a representation of a function? Explain.

2. Which of the following relations represent a function? Explain.

3. Which of the following relations represent a function? Explain.

a)
b)
c)

1.

This is a function because there is one output for every input. In other words, if you think of these points as coordinate points , there is only one value for given for every value of .

2. a)

Since the vertical line hit the graph more than once (indicated by the two green circles), the graph does not represent a function.
b)
Since the vertical line hit the graph only once (indicated by the one green dot), the graph does represent a function.

3. a)

This is a function because there is only one output for a given input.
b)
Since the vertical line hit the graph more than once (indicated by the three blue circles), the graph does not represent a function.
c)
Since the vertical line hit the graph only once (indicated by the one blue dot), the graph does represent a function.

Practice

Determine whether or not each relation is a function. Explain your reasoning.

1. .

1. .

1. .

1. .

1. .

Which of the following relations represent a function? Explain.

1. .
1. .
1. .
1. .
1. .

Which of the following relations represent a function? Explain.

Vocabulary Language: English

Function

Function

A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.
Relation

Relation

A relation is any set of ordered pairs $(x, y)$. A relation can have more than one output for a given input.
Vertical Line Test

Vertical Line Test

The vertical line test says that if a vertical line drawn anywhere through the graph of a relation intersects the relation in more than one location, then the relation is not a function.