Determine whether each relation defines a function, and give the domain and range.

Verified step by step guidance

1. Understand the definition of a function: A relation defines a function if every input (or x-value) corresponds to exactly one output (or y-value). This means no x-value is paired with more than one y-value.
2. Examine the given relation by listing all the input-output pairs. Identify if any input (x-value) repeats with different outputs (y-values). If it does, the relation is not a function.
3. Determine the domain by collecting all the unique input values (x-values) from the relation. The domain is the set of all these x-values.
4. Determine the range by collecting all the unique output values (y-values) from the relation. The range is the set of all these y-values.
5. Summarize your findings: state whether the relation is a function based on the first step, and clearly list the domain and range sets using set notation, for example, \(\{x_1, x_2, \ldots\}\) for domain and \(\{y_1, y_2, \ldots\}\) for range.

Key Concepts

Definition of a Function

A function is a relation where each input (domain element) is paired with exactly one output (range element). This means no input value can correspond to more than one output. Understanding this helps determine if a given relation qualifies as a function.

Domain of a Relation

The domain is the set of all possible input values in a relation. Identifying the domain involves listing all unique first elements from the ordered pairs or the input values from the relation, which is essential for describing the function's input scope.

Range of a Relation

The range is the set of all possible output values in a relation. It consists of all unique second elements from the ordered pairs or output values. Determining the range helps understand the possible outputs the function or relation can produce.