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

Verified step by step guidance

1. Step 1: Identify the domain and range from the diagram. The domain consists of the input values (left set): \(\{8, 14, 21, 27, 1\}\), and the range consists of the output values (right set): \(\{5, 13, 30\}\).
2. Step 2: Determine if the relation defines a function by checking if each element in the domain maps to exactly one element in the range. In the diagram, each domain value has exactly one arrow pointing to a single range value.
3. Step 3: Notice that the domain values 8 and 14 both map to 5, which is allowed in a function because different inputs can have the same output.
4. Step 4: Since no domain value maps to more than one range value, the relation is a function.
5. Step 5: Summarize the domain and range: Domain is \(\{8, 14, 21, 27, 1\}\) and Range is \(\{5, 13, 30\}\).

Key Concepts

Function Definition

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

Domain and Range

The domain is the set of all possible inputs in a relation, while the range is the set of all possible outputs. Identifying these sets from a relation or mapping diagram is essential for analyzing the behavior and scope of the function.

Mapping Diagrams

Mapping diagrams visually represent relations by showing arrows from elements in the domain to elements in the range. They help easily identify if each domain element maps to one or multiple range elements, aiding in function verification.