Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.

Verified step by step guidance

1. Identify the relation: A relation is a set of ordered pairs.
2. Determine if the relation is a function: A relation is a function if each input (x-value) corresponds to exactly one output (y-value).
3. Check for repeated x-values: If any x-value is repeated with a different y-value, the relation is not a function.
4. Find the domain: The domain is the set of all possible x-values (inputs) in the relation.
5. Find the range: The range is the set of all possible y-values (outputs) in the relation.

Key Concepts

Function Definition

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 for every x-value in the domain, there is a unique y-value in the range. Understanding this definition is crucial for determining whether a given relation qualifies as a function.

Domain and Range

The domain of a function is the complete set of possible input values (x-values) that can be used in the function, while the range is the set of possible output values (y-values) that result from those inputs. Identifying the domain and range helps in understanding the behavior of the function and its limitations.

Vertical Line Test

The vertical line test is a visual method used to determine if a relation is a function. If any vertical line drawn on the graph of the relation intersects it at more than one point, the relation is not a function. This test provides a quick way to assess the functional nature of a relation graphically.