Basic Definitions

The horizontal coordinates of a set of ordered pairs constitutes the domain of a relation, while the range is determined by the vertical coordinates. A relation in which none of the domain numbers appear more than once is called a “function.” If two functions have a common domain, they can be acted upon arithmetically.

Following are definitions with which you should be familiar as you work with relations and functions.

A relation is a set of ordered pairs that can be represented by a diagram, graph, or sentence. Figure 1 shows several examples of relations.

Figure 1 Examples of relations.

Example 1: Find the domain and range of item (a) in Figure 1 .

Example 2: Find the domain and range in item (b) of Figure 1 .

Example 3: Find the domain and range in item (c) of Figure 1 .

Example 4: Find the domain and range in item (d) of Figure 1 .

The domain and range cannot be listed as in the previous examples. In order to visualize the domain, imagine each point of the graph going vertically to the x-axis. The points on the x-axis become the domain (see Figure 2 ).

Figure 2 The domain is horizontal.

To visualize the range, have all the points move horizontally to the y-axis. The points on the y-axis become the range (see Figure 3 ).

Figure 3 The range is vertical.

Example 5: Find the domain and range in item (e) of Figure 1 .

Since any value for x produces a y-value and any value for y produces an x-value,

Example 6: Which of the examples given in Figure 1 are functions?

1. {(4,1),(3,2),(−1,6)

   This is a function, since domain values are not repeated.

2. The example, shown in Figure 4 , is not a function.
   
   

   Figure 4 This is not a function.

   The diagram represents the set of ordered pairs {(1,4), (1,5), (3,4), (2,5)} and (1,4) and (1,5) repeat domain values.

3. The example, shown in Figure 5 , is not a function.
   
   

   Figure 5 Like Figure 4 , this is also not a function.

   The graph represents the set of ordered pairs {(−2,−2), (−1,−2), (−1,1), (0,−1), (1,1), (2,3)}, and (−1,−2) and (−1,1) repeat domain values.

   Notice that the vertical line x = −1 would pass through two points of the graph, as shown in Figure 6 .
   
   

   Figure 6 Functions have no repeated x values.

   A vertical line that passes through (intersects) a graph in more than one point indicates ordered pairs that have repeated domain values and eliminates the relation from being called a function. This test for functions is called the vertical line test.

4. The example shown in Figure 7 is also not a function. It fails the vertical line test.
   
   

   Figure 7 This fails the vertical line test and is, therefore, not a function.

5. y = 2 x + 3.

   This is a function, since domain values are not repeated. When y = 2 x + 3 is graphed, it can be seen that the graph passes the vertical line test.

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