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# Relations

A relation is a set of ordered pairs that defines the relationship between two given sets of information. If two sets of information are available, a relationship defines a map or a connection between them.

## Relations and ordered pairs

An ordered pair is a pair of values written in a particular order. It's common to see ordered pairs represented as $\left(x,y\right)$ with x being the x-variable and y being the y-variable. In numeric form, you might see an ordered pair like $\left(5,2\right)$ or $\left(-3,4\right)$ . You could also see names and whole numbers like (James, 5).

When it comes to relations, your usual goal is to look at the relationship between the set of ordered pairs. You might have a rule that x is related to y if and only if x is less than y. In this case, an ordered pair like $\left(2,3\right)$ would be related While $\left(3,2\right)$ would not be related.

## What are the domain and range of a relation?

The domain and range help you identify values that go into a relation (domain) as well as the values that emerge from a relation (range). The domain represents all of the x-values in a set of ordered pairs, while the range represents all y-values in a set of ordered pairs.

Take a look at the following ordered pairs in (x,y) form. Can you determine which values represent the domain and which represent the range?

$\left(0,2\right),\left(-3,5\right),\left(1,-4\right),\left(5,7\right),\left(3,2\right)$

Since the domain represents all of the x-values, you'll pull all of the x-values to create the list of values in the domain:

$0,-3,1,5,3$

It's common to place the domain values in numerical order:

$-3,0,1,3,5$

Place brackets around these x-values, and you have your domain set:

$\mathrm{Domain}=\left\{-3,0,1,3,5\right\}$

Now, let's look for the range of the ordered pairs:

$\left(0,2\right),\left(-3,5\right),\left(1,-4\right),\left(5,7\right),\left(3,2\right)$

We'll pull all the y-values from the ordered pairs:

$2,5,-4,7,2$

Let's place them in numerical order:

$-4,2,2,5,7$

It's important to note that you should not repeat values in domain and range sets. As you can see, when placing the y-values in numerical order, 2 is easily located twice among the values. We'll only keep one 2 for the range set:

$\mathrm{Range}=\left\{-4,2,5,7\right\}$

Here's how this will look when all put together (ordered pairs have been placed in brackets to show they form a set):

$\mathrm{Ordered pairs}=\left\{\left(0,2\right),\left(-3,5\right),\left(1,-4\right),\left(5,7\right),\left(3,2\right)\right\}$

$\mathrm{Domain}=\left\{-3,0,1,3,5\right\}$

$\mathrm{Range}=\left\{-4,2,5,7\right\}$

## Is a function a relation?

A function is a special type of relation. It is a set of ordered pairs in which no two different ordered pairs have the same x-coordinates. For a relation to be a function, every input value (x-value) must have only one unique output value (y-value).

How can you tell that every input value only has one output value? Let's take another look at these ordered pairs:

$\left(0,2\right),\left(-3,5\right),\left(1,-4\right),\left(5,7\right),\left(3,2\right)$

You can see that every input (x) value has only one output (y) value. This means the relation is a function.

But what if the ordered pairs look like this?

$\left(0,2\right),\left(3,5\right),\left(1,-4\right),\left(5,7\right),\left(3,2\right)$

The input value 3 has outputs of 5 and 2. In this case, the relation is not a function.

It's important to note that while a function cannot have an input that has more than one output, two different inputs can have the same output:

$\left(0,2\right),\left(-3,5\right),\left(1,-4\right),\left(5,7\right),\left(3,2\right)$

A relation can also be a function if an input has the same output more than once in the same relation:

$\left(0,2\right),\left(-3,5\right),\left(1,-4\right),\left(5,7\right),\left(3,2\right)$

## Input-output tables

The input-output table is another way to determine whether a relation is a function. As mentioned previously, ordered pairs $\left(x,y\right)$ are known as input $\left(x\right)$ values and output $\left(y\right)$ values. To see if a relation is a function, you can create a 2-column table, label the left side "Input" and the right side "Output," then list the x-values and y-values as ordered pairs.

Here is an example using the relation $\left(1,7\right),\left(-2,6\right),\left(0,3\right),\left(1,4\right),\left(0,3\right)$ :

 Input (x) Output (y) 1 7 -2 6 0 3 1 4 0 3

In this example, input 1 has an output of 7 and an output of 4. Since this relation has an input with more than one output, it is not a function.

Now, let's look at the input-output table for the relation $\left(5,2\right),\left(0,4\right),\left(-4,3\right),\left(1,4\right),\left(3,1\right)$ :

 Input Output 5 2 0 4 -4 3 1 4 3 1

Every input in this relation has only one output. This means this relation is a function.

## Mapping diagrams

A mapping diagram can also help you determine if a relation is a function. When creating a mapping diagram, you use the domain and range.

Let's say that you have a relation with the ordered pairs $\left(-1,5\right),\left(0,3\right),\left(1,5\right),\left(2,11\right),\left(3,21\right)$ . Your domain is $\left\{-1,0,1,2,3\right\}$ and your range is $\left\{3,5,11,21\right\}$ . To see how these elements are paired, you'll draw the mapping diagram, which consists of two parallel columns. The first column represents the domain and the other column represents the range.

Lines (or arrows) are drawn from the domain to the range to showcase the relationship between elements:

The mapping diagram above shows that the second element of the range associates with more than one element in the domain. This diagram represents many-to-one mapping. In this case, this relation is a function.

If you have elements in the domain column that associate with more than one element in the range column, this is called one-to-many mapping:

Since input a has more than one output, this mapping diagram shows that the relation is not a function.

If a mapping diagram shows that each element of the range is paired with exactly one element of the domain, this is called one-to-one mapping:

This mapping diagram represents a function.

## The vertical line test

The vertical line test is another great way to determine if a relation is a function. The vertical line test says that a graph represents a function if and only if all vertical lines intersect the graph at most once.

Here is an example of a graph that represents a function:

All vertical lines intersect the graph exactly once, and doesn't intersect at all.

Now, take a look at this graph:

The vertical line $x=1$ intersects in three places, which means this graph does not represent a function.

## Practice questions on relations

a. What are the domain and range of the following relation: $\left\{\left(2,1\right),\left(-3,7\right),\left(-1,0\right),\left(5,1\right)\right\}$ ?

$\mathrm{Domain}=\left\{-3,-1,2,5\right\}$

$\mathrm{Range}=\left\{0,1,7\right\}$

b. Does this relation represent a function: $\left\{\left(2,1\right),\left(-3,7\right),\left(-1,0\right),\left(5,1\right)\right\}$ ?

Yes, because there is no input with more than one output.

c. Does the input-output table for this relation represent a function?

 Input Output 5 2 0 4 -1 6 5 3 0 4

No, because the input value 5 has more than one output.

d. What is one-to-one mapping?

When a mapping diagram shows that each element of the domain is paired with exactly one element of the range.

e. According to the vertical line test, when does a graph represents a function?

If and only if all vertical lines intersect the graph at most once.

f. What do input and output values represent?

Input = x-values and Output = y-values

g. Is a relation always a function?

No, a relation is only a function if each input value has only one output value.

Functions

## Get help learning about relations

When learning about relations, your student will go over a lot of concepts, including ordered pairs, domain and range, and functions. Relations can be fun, but they can also be challenging, particularly when trying to find the domain and range or determine if a relation is a function. It's essential that students have a firm grasp of all of the foundational skills behind relations. If your student feels confused by any of these concepts, working with a tutor can make a significant difference. Even if they're not struggling with relations and just want help keeping up or preparing for a test, tutoring can help. Learn more about tutoring and its benefits by contacting the Educational Directors at Varsity Tutors today.

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