Final answer:
A function in mathematics is a relationship where each input is associated with a single, unique output. Relations that assign multiple outputs to the same input or diverge are not considered functions. The analysis of a relation to ascertain if it is a function includes checking that each input (domain) maps to exactly one output (range).
Step-by-step explanation:
In mathematics, a function is defined as a relationship between two sets that assigns to each element of the first set (commonly referred to as the domain) exactly one element of the second set (the range). To determine if a given relation is a function, one must check whether each input value has a unique output value. A relation that assigns multiple output values to a single input is not considered a function.
For instance, if a professor's name is Adam Smith, we might describe the relationship as Professor = Adam Smith, signifying a one-to-one relationship, which could be expressed as a function. Similarly, if we have a group of friends consisting of Bob, Shawn, and Margaret, describing the set of friends can also be akin to a function if each person is counted once.
However, certain relations, like those described in the student's question where a relation is double-valued or diverges, cannot be functions, as they do not satisfy the requirement of each input being associated with a single, unique output.