A relation qualifies as a function if it adheres to a specific rule: each input value must correspond to exactly one output value.
Understanding Relations and Functions
A relation, in simple terms, is a set of ordered pairs. These pairs connect two sets of values. For instance, {(1, 2), (2, 4), (3, 6)} is a relation, where the first number in each pair is considered the 'input' (often denoted as 'x') and the second number is the 'output' (often denoted as 'y').
A function, however, is a special type of relation with a specific requirement.
Key Rule for a Function
The core rule for a relation to be considered a function, as highlighted by the reference, is:
- Every input (x-value) must be associated with only one output (y-value).
This means that you cannot have a single input value that maps to multiple different output values. If this happens, the relation is not a function.
Practical Examples
Let's illustrate with a few examples:
- Example 1: Function
{(1, 2), (2, 4), (3, 6)}- Here, each input (1, 2, 3) has only one unique output (2, 4, 6). This is a function.
- Example 2: Not a Function
{(1, 2), (1, 3), (2, 4)}- Notice that input '1' is associated with two outputs: '2' and '3'. This violates the rule, so it is not a function.
- Example 3: Function
{(1, 5), (2, 5), (3, 5)} - Each input (1, 2, 3) has a unique output of '5', which is perfectly acceptable. Multiple different inputs can map to the same single output and still be considered a function.
In Summary
To check if a relation is a function, always ensure that no single input value (x) is paired with more than one output value (y). This is crucial for a relation to qualify as a function. As the reference states, a function is a specific kind of relation where every X-value should be associated with only one y-value.
