Determine whether each relation is a function. Give the domain and range for each relation. {(3, 4), (3, 5), (4, 4), (4, 5)}

Verified step by step guidance

1. Step 1: Understand the definition of a function. A relation is a function if each input (x-value) is associated with exactly one output (y-value). In other words, no x-value can be repeated with different y-values.
2. Step 2: Examine the given relation: {(3, 4), (3, 5), (4, 4), (4, 5)}. Identify the x-values (inputs) and check if any x-value is repeated with different y-values.
3. Step 3: Notice that the x-value '3' is paired with both '4' and '5', and the x-value '4' is paired with both '4' and '5'. This indicates that the relation is not a function because the same x-value is associated with multiple y-values.
4. Step 4: Determine the domain of the relation. The domain is the set of all x-values in the relation. For this relation, the domain is {3, 4}.
5. Step 5: Determine the range of the relation. The range is the set of all y-values in the relation. For this relation, the range is {4, 5}.

Key Concepts

Definition of a Function

A function is a specific type of relation where each input (or domain element) is associated with exactly one output (or range element). This means that no two ordered pairs can have the same first element with different second elements. Understanding this definition is crucial for determining whether a given relation qualifies as a function.

Domain and Range

The domain of a relation is the set of all possible first elements (inputs) from the ordered pairs, while the range is the set of all possible second elements (outputs). Identifying the domain and range helps in understanding the scope of the relation and is essential for analyzing its properties, including whether it is a function.

Ordered Pairs

Ordered pairs are pairs of elements written in the form (x, y), where 'x' is the first element and 'y' is the second element. In the context of relations, the arrangement of these pairs is significant, as it determines the relationship between the inputs and outputs. Analyzing the ordered pairs in a relation is key to assessing its function status.