Domain and Range

A relation is a set of ordered pairs. In each pair, the first element comes from a set called the domain, the second from a set called the range.

Example 1

Domain:  \{\text{Harry Potter, Hermione Granger, Ron Weasley, Draco Malfoy, Cedric Diggory, Luna Lovegood}\}

Range: \{\text{Gryffindor, Hufflepuff, Ravenclaw, Slytherin}\}

Now let a represent an element from the domain, and b represent an element form the range. We can define the relation:

a is related to b if a is a member of the house of b.

This generates the following relation:

    \[\{\text{(Harry Potter, Gryffindor),(Hermione Granger, Gryffindor),(Ron Weasley, Gryffindor),}\]

    \[\text{(Draco Malfoy, Slytherin),(Cedric Diggory, Hufflepuff),(Luna Lovegood, Ravenclaw)}\}\]

In this relation, there are six elements in the domain; four in the range and six ordered pairs in the relation.

A relation is a broad concept, and can have some meaning that is recognisable or can seem quite random.

Example 2

    \[\{(3,5), (0,7), (1,8), (-3,8)\}\]

It seems a little random, and it is. However it fits the definition of a set of ordered pairs, and so it is a relation. In this case, we have the following:

Domain: \{-3, 0, 1, 3\}

Range: \{5, 7, 8\}

There is no general way to define how they are related, so the relation itself is the definition. In this example a total of four elements in the domain, in the range, and in the relation.

Example 3

Domain: All the real numbers

Range: Not sure yet, it depends on what the relation is …

Let x be a member of the domain. Let y be a member of the range. Let’s define the relation as follows:

x is related to y if x^2=y

This definition generates the following ordered pairs:

    \[\{(3, 9), (5, 25), (1, 1), (0,0), (-5, 25) \dots \}\]

Notice that we can square any number, but when we do, the answer is always positive. Therefore, the range includes all numbers greater than or equal to zero, but no negative numbers. In conclusion, we have:

The relation: ordered pairs (x, y) where y=x^2

Domain: x is any real number

Range: y \geq 0

Example 4

We can read the domain and range of a relation from a graph.

For the domain, we’re looking at the horizontal axis. In this case, the x axis. We’re looking for the minimum value of x and the maximum value of x included on the set of green points that makes up the curve.

For the range, we’re looking at the vertical axis. We’re looking for the minimum and the maximum value of y on the green curve.

Domain: -8 \leq x \leq 5

Range:  -7.7 \leq y \leq 11.2

Notice that we can read the domain from the endpoints, but the minimum and maximum values of y could appear anywhere in between the endpoints.

Example 5

In this graph of a relation, the max and min values of both x and y occur at the endpoints.

Domain: -7 \leq x \leq 8

Range:  -13.6 \leq y \leq 16.4

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