91Ó°ÊÓ

To understand whether a relation is a function, you first need to grasp the difference between the two concepts. A relation is any set of ordered pairs. For example, \(\{(0.5,7),(0,7),(1,7),(9,7)\}\) is a relation because it consists of pairs of inputs and outputs.

However, for a relation to be classified as a function, it must meet a specific criterion: each input (or 'x' value) must pair with exactly one output (or 'y' value). In other words, no 'x' value is allowed to have more than one 'y' value.

Think of a function like a vending machine. You press a button (input), and you get exactly one snack (output). If pressing the same button could give you different snacks each time, that would not be a 'function' in the mathematical sense. Thus, ensuring that each 'x' corresponds to one and only one 'y' is key to calling a relation a function.

The terms 'domain' and 'range' are essential when discussing functions and relations. The domain is the set of all possible input values (x-values) in a relation or function. On the other hand, the range is the set of all possible output values (y-values).

In our exercise, the domain includes the x-values 0.5, 0, 1, and 9. These are the inputs. The range includes only one unique y-value: 7.

Understanding the domain and range helps you determine whether a relation is a function. If any x-value from the domain maps to more than one y-value in the range, the relation is not a function. In our given example, because each x-value maps to exactly one y-value (7), the relation qualifies as a function.

To confirm if a relation is a function, checking for unique inputs is crucial. Unique inputs mean that each 'x' value should appear only once in the set of ordered pairs.

For instance, in the relation \(\{(0.5,7),(0,7),(1,7),(9,7)\}\), the x-values are 0.5, 0, 1, and 9. These values are all unique and different from each other, making it simpler to verify that the relation is a function.

Always remember that the key reason for verifying unique inputs is to ensure that each x-value links to only one y-value. If an x-value repeats, with different y-values assigned to these repeats, the relation fails the function test. This is because a single input giving multiple outputs contradicts the definition of a function. In our case, since all inputs are unique and none of them repeat, we confidently conclude that the relation is indeed a function.