Problem 77 - College Algebra

Question

Consider the relation for which the domain represents the ten longest-running series and the range represents the number of seasons the series ran. Is this relation a function? Explain your answer.

Step-by-Step Solution

Answer

Yes, the given relation is a function because each of the ten longest-running series corresponds to exactly one number of seasons it ran. This fits the definition of a function, where each input has exactly one output.

Step 1: Understanding the relation

In the given relation, each long-running series is linked with a specific number of seasons it ran. This pairing creates a unique relation between each series and the number of its seasons.

Step 2: Verifying the definition of a function

A function must satisfy the condition that every input corresponds to exactly one output. Here, each input, i.e., each series, has one output - the number of seasons it ran. Therefore, every series from the ten longest-running ones is related to exactly one number from the number of seasons.

Step 3: Conclusion

As per the definition of a function, this given relation satisfies the condition. Therefore, this relation can be considered a function.

Key Concepts

relations

In mathematics, a relation refers to a connection or association between two sets of information. This can be thought of as a link between the domain (the input or starting points) and the range (the outcomes or end points). In our exercise, the relation involves the ten longest-running TV series as the domain and the number of seasons they ran as the range. This type of relation expresses which series is connected to how many seasons.

Relations don't necessarily have to follow a strict pairing rule. Multiple series might have run for the same number of seasons. However, each series is still distinctly associated with its own specific count of seasons. So, even when multiple series share the same number, as long as there's a clear way of pairing between each series and the seasons it ran, the relation holds.

domain and range

The concepts of domain and range are vital in understanding relations and functions. The **domain** is the set of starting points - in our case, the ten longest-running series. The **range** is the set of outcomes - the number of seasons each series ran for.

Domains and ranges are crucial because they tell us the limits and possibilities of a relation or function. When dealing with TV series and their seasons, the domain provides all the possible series under consideration. The range gives us the set of seasons these series might span. It's important that each item in the domain (each series) maps to a specific item in the range (a definite number of seasons) to form what's called a function, which we'll soon explore.

function definition

A function is a special type of relation where each element from the domain is paired with exactly one element in the range. This characteristic is what differentiates functions from general relations.

Here, each TV series (element of the domain) is linked to one particular number of seasons (element of the range). This means that the relation fits the definition of a function because every series correlates to one unique number of seasons. Functions help us provide clarity and predictability. They let us be sure that whenever we choose a series, we can predict precisely how many seasons it ran by checking our map of series to seasons.