Final answer:
A relation is a set of ordered pairs possibly with multiple outputs for a single input, while a function is a special kind of relation with each input uniquely paired to one output, useful for modeling economic relationships or determining connections between variables such as exam scores.
Step-by-step explanation:
A relation in mathematics is a set of ordered pairs where the first elements are considered inputs and the second elements are outputs. It can show that one thing is related to another, without the necessity of a one-to-one relationship. This is like saying Friends = Bob + Shawn + Margaret, where 'Friends' is related to several people.
In contrast, a function is a special type of relation where each input is related to exactly one output. This means for any given input (x), there's only one output (y). This is similar to the statement Professor = Adam Smith, indicating a unique association between the term 'Professor' and the individual 'Adam Smith'.
Examining the relationship between two variables is essential for professionals who often seek to understand if there's a connection and how strong it is. For example, one might want to know if there is a relationship between the grades on a second math exam and those on the final exam. Such a relationship, if it exists, can often be modeled using linear equations.