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# Vertical Line Test

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Practice Vertical Line Test
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Relations and Functions

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

$& x \qquad 5 \qquad \ 10 \qquad 15 \qquad 10 \qquad \ 5 \qquad \ 0\\& y \qquad 12 \qquad 25 \qquad 37 \qquad 55 \qquad 72 \qquad 0$

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 $y$ for every value of $x$ .

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

Table A
$x$ $y$
0 4
1 7
2 7
3 6
Table B
$x$ $y$
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.

$x$ $y$
$-3.5$ $-3.6$
$-1$ $-1$
4 3.6
7.8 7.2

Solution:

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

#### 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

$& x \qquad 5 \qquad \ 10 \qquad 15 \qquad 10 \qquad \ 5 \qquad \ 0\\& y \qquad 12 \qquad 25 \qquad 37 \qquad 55 \qquad 72 \qquad 0$

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 $x$ , there is only one value for $y$ .
Relation
A relation is any set of ordered pairs $(x, y)$ . 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.

$s = \{(1, 2), (2, 2), (3, 2), (4, 2)\}$

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

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

a) $& x \qquad 2 \qquad 4 \qquad \ 6 \qquad \ 8 \qquad \ 10 \qquad 12\\& y \qquad 3 \qquad 7 \qquad 11 \qquad 15 \qquad 19 \qquad 23$
b)
c)

1. $s=\{(1,2),(2,2),(3,2),(4,2)\}$

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

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) $& x \qquad 2 \qquad 4 \qquad \ 6 \qquad \ 8 \qquad \ 10 \qquad 12\\& y \qquad 3 \qquad 7 \qquad 11 \qquad 15 \qquad 19 \qquad 23$

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. .
$& X \qquad 2 \qquad \quad 3 \qquad 2 \qquad \quad \ 5\\& Y \qquad 3 \qquad -1 \qquad 5 \qquad -4$
1. .
$& X \qquad 4 \qquad 2 \qquad \quad 6 \qquad -1\\& Y \qquad 2 \qquad 4 \qquad -3 \qquad \quad 5$
1. .
$& X \qquad 1 \qquad 2 \qquad 3 \qquad 4\\& Y \qquad 5 \qquad 8 \qquad 5 \qquad 8$
1. .
$& X \qquad -6 \qquad -5 \qquad -4 \qquad -3\\& Y \qquad \quad 4 \qquad \quad \ 4 \qquad \quad \ 4 \qquad \quad \ 4$
1. .
$& X \qquad -2 \qquad 0 \qquad -2 \qquad 4\\& Y \qquad \quad 6 \qquad \ 4 \qquad \quad \ 4 \qquad 6$

Which of the following relations represent a function? Explain.

1. $s=\{(-3,3),(-2,-2),(-1,-1),(0,0),(1,1),(2,2),(3,3)\}$
2. $s=\{(1,1),(1,2),(1,3),(1,4),(1,5)\}$
3. $s=\{(1,1),(2,1),(3,1),(4,1),(5,1)\}$
4. $s=\{(-3,9),(-2,4),(-1,1),(1,1),(2,4)\}$
5. $s=\{(3,-3),(2,-2),(1,-1),(0,0),(-1,1),(-2,2)\}$