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# Determining if a Relation is a Function

Concept

## Relation

A relation, or relationship, shows how two quantities are related — it pairs inputs with outputs. Often, a relation is thought of as a set of $(x,y)$ coordinates.
Method

## Representing a Relation

A relation can be represented in various ways, of which a few are shown here.

Method

### Set of Coordinates

When a relation is given as a set of coordinates, such as $\{(0,1),$ $(\text{-}1,8),$ $(2,4)\},$ the $x$-coordinates represent the inputs, and the $y$-coordinates represent the corresponding outputs. In this example, the input $0$ is paired — or mapped — to the output $1,$ the input $\text{-} 1$ is mapped to the output $8,$ and the input $2$ is mapped to the output $4.$

Method

### Table of Values

A table where one row gives the input and another row gives the output, is a representation of a relation.

 Input ($x$) Output ($y$) $\text{-} 2$ $1$ $5$ $3$ $2$ $1$

The relation given by the above table is the same as $\{(\text{-} 2, 3),$ $(1, 2),$ $(5, 1)\}.$

Method

### Graph

Instead of listing the set of coordinates, they can be plotted in a coordinate plane. The $x$-coordinates are the inputs, and the $y$-coordinates are the corresponding outputs.

The graph visualizes $\{(\text{-} 3, \text{-} 2),$ $(1, \text{-} 1),$ $(1, 3),$ $(4, \text{-} 1)\}.$

Method

### Mapping Diagram

A mapping diagram is a diagram that lists the inputs in one column, and the outputs in another column. Arrows are used to indicate which outputs correspond to which inputs.

This mapping diagram shows $\{(\text{-} 2, \text{-} 2),$ $(\text{-} 1, 0),$ $(0, 2),$ $(2, \text{-} 2)\}.$

Method

### Rule

The relation between inputs and outputs can also be defined using a rule. One such rule is $y = 2x + 1.$

Replacing $x$ with any input gives the value of the corresponding output, $y.$ For instance, the input $x = 1$ yields the output $y = 3.$ Thus, this relation maps the input $1$ to the output $3.$
Concept

## Function

A function is a relation in which each input value corresponds to exactly one output value. If $x$ represents the inputs and $y$ the outputs of a function, it is often said that $y$ is a function of $x."$ Depending on the way in which the relation is represented, deciding if it is a function can be done in different ways. If the relation is given as a set of coordinates, representing it with a mapping diagram can be helpful.
Exercise

Represent the following relation with a mapping diagram and determine whether it is a function. $\{(1,2),(2,1),(\text{-} 2,\text{-} 1),(3,\text{-} 1),(2,2)\}$

Solution

To create the mapping diagram we'll start by listing the inputs in a column to the left, and the outputs in a column to the right. The inputs of the relation are $1, 2, \text{-} 2,$ and $3,$ and the outputs are $2, 1,$ and $\text{-} 1.$

We can now add the arrows mapping the inputs to their corresponding outputs. There should be an arrow from input $1$ to output $2,$ from input $2$ to output $1,$ etc.

For a relation to be a function, every input must correspond with exactly one output. Here, we can see that the input value $2$ has two possible outputs, $2$ and $1.$ Thus, this relation is not a function.

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Method

## Vertical Line Test

Relations that are not functions have the same specific characteristic when graphed. When a relation is not a function, it is because there are multiple $y$-values produced by the same $x$-value. Therefore, the graph would show at least two points directly above the other. Consider the following graphs.

This fact gives way to a test that can be done to determine if a relation is a function by looking at its graph. This test is called the Vertical Line Test, and is performed by moving an imaginary vertical line across the graph. If this line intersects the graph more than once at any given time, the relation is not a function. If this is not the case, the relation is a function.

Perform test

Change relation

Note that an assumption about the graph's behavior must be made. It must be assumed that the graph continues on in the same way, without any significant changes, outside of the coordinate plane. If not, it could never be concluded from a graph that a relation is a function.
Exercise

Is the graphed relation a function?

Solution

Since the relation is given as a graph, we can use the Vertical Line Test to figure out if it's a function. Moving an imaginary vertical line across the graph, we can see that it never intersects the graph more than once at any given time.

Thus, under the necessary assumption that the graph behaves in a similar fashion outside of the visible region, the relation is a function.

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