Problem 10 - College Algebra Essentials

Question

Determine whether each relation is a function. Give the domain and range for each relation. $$ \\{(4,1),(5,1),(6,1)\\} $$

Step-by-Step Solution

Answer

The given relation is a function with the domain as the set \{4,5,6\} and the range as the set \{1\}.

Step 1: Determine if the Relation is a Function

In this case, the ordered pairs given are \((4,1), (5,1), (6,1)\). In all these pairs, the first element (x-value) does not repeat. Therefore, every element in the domain corresponds to exactly one element in the range. As such, this relation qualifies as a function.

Step 2: Identify the Domain

The domain of a function is the set of all possible x-values. In these ordered pairs, the x-values are 4, 5, and 6. So, the domain is the set \{4,5,6\}.

Step 3: Identify the Range

The range of a function is the set of all possible y-values. In these ordered pairs, the y-value is 1 for all pairs. So, the range is the set \{1\}.

Key Concepts

Domain

When considering any mathematical function, the domain is one of the primary aspects to understand. The domain of a function is simply the set of all possible input values, often represented by the x-values in ordered pairs. It's essential because it defines what inputs the function can accept and work with.

For the relation given in the exercise \( \{(4,1),(5,1),(6,1)\} \), the domain includes the values \( \{4,5,6\} \). Each element in the domain corresponds to a unique output value or y-value in the context of this relation.

• The domain is integral for defining the type of function.
• It impacts the function's potential graphs and behaviors.
• In simple cases like linear functions, identifying domain values is usually straightforward as shown in this exercise.

Understanding the domain is crucial not only for evaluating functions but also for broader mathematical applications, determining how functions behave across different sets of numbers.

Range

Just as the domain refers to input values, the range of a function describes the possible output values. In terms of ordered pairs, these are the y-values.

For the particular relation \( \{(4,1), (5,1), (6,1)\} \), the range is restricted to the value \( \{1\} \). This is because, regardless of the input (4, 5, or 6), the output remains constant at 1. Each x-value in the domain leads directly to the same y-value.

• The range defines the actual limits of the function’s outputs.
• It can be used to understand the level of variability or constancy of the function.
• For more complex functions, identifying the range may require deeper analysis or visualization through graphs or tables.

The range paints a clear picture of what kind of outputs a function can produce, helping in predictions or interpretations of real-world scenarios where functions come into play.

Relation

In mathematics, a relation is a set of ordered pairs. It denotes a specific way to connect elements from two sets. Relations give rise to both simple and intricate structures, but it's their specialized cases called functions that are particularly noteworthy.

Within the context of the exercise with the set \( \{(4,1), (5,1), (6,1)\} \), we can classify this relation as a function because no x-value is repeated, and each one corresponds to at least one y-value. In simpler terms, there’s a unique y-value for each x-value.

• A function is a special type of relation where each input (x-value) relates to exactly one output (y-value).
• Understanding whether a relation is a function is crucial in determining how systems and equations behave.
• Functions facilitate predicting outcomes, making them particularly useful in numerous mathematical and real-life applications.

Grasping the concept of relations and their progression into functions lays the groundwork for deeper mathematical exploration and problem-solving.