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.
![\[(\text{element from the domain},\text{element from the range})\]](https://tentotwelvemath.com/wp-content/ql-cache/quicklatex.com-184f191b4428d780331d941f278ccaa9_l3.png "Rendered by QuickLaTeX.com")
Example 1
Every student at Hogwarts’ School is placed in a house by the sorting hat. Let the domain be the set of students attending Hogwarts. Let the range be the set of four houses.
Domain: 
Range: 
We can define the relation:
a student is related to a house if the student is a member of the house.
This definition generates these ordered pairs:
![\[\{\text{(Harry, Gryffindor),(Hermione, Gryffindor),(Ron, Gryffindor),}\]](https://tentotwelvemath.com/wp-content/ql-cache/quicklatex.com-ae0d65e595c275dca5e108c1fb8c7215_l3.png "Rendered by QuickLaTeX.com")
![\[\text{(Draco, Slytherin),(Cedric, Hufflepuff),(Luna, Ravenclaw)\dots}\}\]](https://tentotwelvemath.com/wp-content/ql-cache/quicklatex.com-de6cbb8b53eae2dedfce495d613b7afd_l3.png "Rendered by QuickLaTeX.com")
A relation is a broad concept. The elements don’t have to be numerical. The remaining examples on this page are numerical.
Example 2
Here is a small, somewhat random, set of ordered pairs:
![\[\{(-3,8), (0,7), (1,8), (3,5)\}\]](https://tentotwelvemath.com/wp-content/ql-cache/quicklatex.com-bf46e8dd21deb5bd9fb528226c8b637a_l3.png "Rendered by QuickLaTeX.com")
The domain is the set of values in the first place.
Domain: 
The range is the set of values in the second place.
Range: 
Example 3
We can read the domain and range of a relation from a graph.
The coordinates of the four points on the graph are:
![\[\{(1, 2), (2, 3), (3, 1), (5,2)\}\]](https://tentotwelvemath.com/wp-content/ql-cache/quicklatex.com-d978973145e480fb71b48c7a077ae0ea_l3.png "Rendered by QuickLaTeX.com")
The domain is the set of 
values 
.
The range is the set of 
values 
. We need only mention the y value ‘2’ once, even though it appears more than once in the set of ordered pairs.
Example 4
The image below shows an eagle. Image source: https://vimarina.ca/latest-news/wildlife-west-coast-expect/
The green rectangle surrounds the eagle. Move the green point to the furthest left feather on the eagle, then to the furthest feather on the right. Move the green point to the highest point on the eagle’s wing. Now move it to the lowest point on the eagle’s tail.
The furthest left, right, the highest and lowest points are all on the boundaries of the rectangle. We can use the rectangle to help determine the domain and range of the set of points on the eagle.
Domain: 
Range: 
Example 5
On the graph below we see a continuous curve. A continuous curve represents infinitely many points, so we can’t list them individually.
Move the green travelling point along the curve.
To identify the domain, look at the horizontal axis. We’re looking for the minimum and maximum value of .
To identify the range, look at the vertical axis. We’re looking for the minimum and the maximum value of .
Domain: 
Range: 
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 6
In some sports, when a ball hits the boundary line of the court it is considered ‘in’ (eg, soccer) in other sports it is considered ‘out’ (eg basketball). In math we use solid lines to show boundaries that are ‘in’, and we use the symbol 
. We use dotted lines to show boundaries that are ‘out’ along with the symbol 
.
Example 7
Similarly with endpoints, we use solid points to show endpoints that are ‘in’ and empty points to show endpoints that are ‘out.’ Note the symbols used in the example below to represent inclusion and exclusion.
Practice writing the domain and range
Use the red dynamic point to find the max/min x coordinates and y coordinates. Click the ‘trace button’ then move the red dynamic point to see the domain/range traced on the x, y axes. Write the domain and the range for each function. Solutions are on the pdf linked below the applet.