How do you find a function in a relation?

To determine if a relation is a function, you check if each x-value is associated with only one y-value.

Identifying Functions from Relations

A relation is any set of ordered pairs. A function is a special type of relation. The key difference, as highlighted in the provided reference, is:

• Function Definition: In a function, each input (x-value) has only one output (y-value).

Method to Identify a Function

The most straightforward method to identify a function from a given relation is to check for repeating x-values:

1. Examine the x-values: Look at all the x-values within the relation.
2. Check for repetitions: Determine if any x-values are repeated.
3. Analyze repeated x-values:
   • If an x-value is not repeated, it meets the function requirement for that value.
   • If an x-value is repeated, check if it is associated with the same y-value each time it appears. If it is, then this x-value meets the function criteria. If the x-value is associated with different y-values, the relation is not a function.

Examples

Here are some examples to illustrate this process:

• Example 1: Function

  The relation {(1, 2), (3, 4), (5, 6)} is a function because no x-value (1, 3, or 5) is repeated.

• Example 2: Not a Function

  The relation {(1, 2), (3, 4), (1, 5)} is not a function because the x-value 1 is associated with two different y-values (2 and 5). This violates the fundamental rule that each x-value can only have one corresponding y-value.

• Example 3: Function (Repeating y-values allowed)

  The relation {(1, 2), (3, 2), (5, 2)} is a function. Although the y-value 2 is repeated, each x-value (1, 3, and 5) is unique, satisfying the function definition.

Practical Insights

• Tables: When a relation is presented in a table, it's easy to visually scan the x-column for repetitions.
• Mapping Diagrams: If a relation is shown as a mapping diagram, ensure that each x-value has only one arrow originating from it.
• Graphs: Use the vertical line test. If a vertical line intersects the graph of the relation at more than one point, then the relation is not a function. This test is effective in visualizing functions.

By carefully examining the x-values and their corresponding y-values, you can accurately determine whether a relation is indeed a function.