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 domain value) corresponds to exactly one output (or range value). This means no input is paired with more than one output.
2. Identify the domain: List all the input values from the given relation. The domain is the set of all these input values.
3. Identify the range: List all the output values from the given relation. The range is the set of all these output values.
4. Check if the relation is a function: For each input in the domain, verify that it is paired with only one output. If any input has multiple outputs, the relation is not a function.
5. Summarize your findings by stating whether the relation is a function, and clearly write down 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 (usually x-values) in a relation. Identifying the domain involves listing all unique inputs that appear in 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 (usually y-values) that result from the inputs in the domain. Finding the range requires collecting all unique outputs associated with the inputs, providing insight into the function's possible outputs.