Problem 2 - College Algebra

Question

determine whether each relation is a function. Give the domain and range for each relation. $$ [(4,5),(6,7),(8,8)] $$

Step-by-Step Solution

Answer

The given relation is a function. Its domain is {4, 6, 8} and range is {5, 7, 8}.

Step 1: Determine if the relation is a function

In this relation, \( [(4,5),(6,7),(8,8)] \), each x-value is unique, which means each element in the domain maps to exactly one element in the range. Thus, this relation is a function.

Step 2: Find the domain

The domain of a function consists of all the first elements (x-values) of the ordered pairs. In this function, the domain is \( {4, 6, 8} \).

Step 3: Find the range

The range of a function consists of all the second elements (y-values) of the ordered pairs. In this function, the range is \( {5, 7, 8} \).

Key Concepts

Domain and Range

In the context of relations and functions, the domain and range are fundamental concepts that every student should master. The domain refers to the set of all possible input values (or "x-values") for a relation or a function. Essentially, it's all the values that we plug into a function. In our example, the domain is represented by the first parts of each ordered pair:

• (4, 5)
• (6, 7)
• (8, 8)

Here, the domain is the set \({4, 6, 8}\).
On the other hand, the range is the set of all possible output values (or "y-values") that result from employment of the domain elements into a relation or function. Continuing with our example, the range comprises the second values of each ordered pair:

• (4, 5)
• (6, 7)
• (8, 8)

This makes the range \({5, 7, 8}\). These concepts help determine how a function behaves and are crucial for understanding the entire scope of a function's potential outputs.

Ordered Pairs

Ordered pairs are a way of representing elements in relations and functions, usually denoted as \(x, y\), where \(x\) is the first element and \(y\) is the second. They are fundamental in expressing relations between sets of data.
The order is vital because the first element gives us the domain value, while the second gives us the range. In our specific example, we have the set of ordered pairs \([ (4, 5), (6, 7), (8, 8) ]\). Each pair is like a little instruction, telling us how to connect each \(x\)-value with its corresponding \(y\)-value.
By examining these pairs, one can infer the function's traits and whether it truly behaves as a function. Remember, if no two ordered pairs have the same \(x\)-value but different \(y\)-values, it's a function! It's the clarity of this ordered structure that allows us to understand how \(x\) and \(y\) relate and interact.

Discrete Mathematics

Discrete mathematics is a branch of mathematics that deals with discrete elements, focusing on the study of countable, distinct, and separable objects. It's like zooming in on mathematics at a granular level.
One of the key areas in discrete mathematics is functions—specifically how they relate to ordered set theory. The study of relations and functions, especially ones represented by ordered pairs, like \([ (4, 5), (6, 7), (8, 8) ]\), falls under this umbrella.
In discrete mathematics, we delve into the analysis of structures like sets, logic, number theory, and graph theory. Each of these elements is essential for recognizing and solving problems accurately in the discrete realm. Understanding discrete mathematics allows us to better frame mathematical functions and relations and makes complex systems more understandable by breaking them down into individually defined parts. With these tools, relations are made manageable, opening up a world of problem solving and computation that is both precise and powerful.