An example of a relation might look like { (2,3), (2,5), (0,1) }. Here we have a relation consisting of three ordered pairs or points on the Cartesian coordinate system. We are familiar with ordered pairs and usually see them denoted as (

*x*,

*y*). Typically the

*x*-value (the first component) will be the independent variable or input and the

*y*-value (the second component) is the dependent variable or output.

The example { (2,3), (2,5), (0,1) } is

**NOT**a function because the

*x*-value 2 is assigned more than one

*y*-value, namely 3 and 5. For every

*x*-value there can be only one

*y*-value. Next we define the domain as the set of

*x*-values and the range as the set of

*y*-values for which the relation is defined.

Example:

**Tip**: When looking at a list of ordered pairs, if there are repeating

*x*-values then the relation is not a function. This usually indicates that there is an input with multiple outputs. (This does not apply to the

*y*-values)

**Determine if the relations are functions. If so state the domain and range.**

Relations can consist of an infinite number of ordered pairs, in which case, making a big list would be impossible. A graph can represent a relation by considering it as a big list of ordered pairs. Each dot on the graph represents an ordered pair (

*x*,

*y*).

Notice that if we can draw a vertical line that intersects the graph twice we will be able to identify one

*x*-value with two corresponding

*y*-values. Therefore, it can not be a function.

Alternatively, if any vertical line crosses the graph only once then it does represent a function.

**Use the vertical line test to determine whether or not the graph represents a function.**

Be prepared to state the domain and range given the graph. Remember to think of the graph as an infinite set of ordered pairs (

*x*,

*y*). From the graph, determine the

*x*-values and

*y*-values. First shade in the domain and range then convert to interval notation.

It should be clear, at this point, that the circle shown is not a function. Nonetheless, we still can determine the domain and range of the relation that it represents.

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