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 pairs of input and output values from the diagram: (8, 5), (14, 5), (21, 13), (27, 30), (1, 30).
2. Determine if each input value is paired with exactly one output value. If any input value is paired with more than one output value, the relation is not a function.
3. In this case, each input value (8, 14, 21, 27, 1) is paired with exactly one output value (5, 5, 13, 30, 30). Therefore, the relation is a function.
4. Identify the domain of the function, which is the set of all input values: {8, 14, 21, 27, 1}.
5. Identify the range of the function, which is the set of all output values: {5, 13, 30}.

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 no two different inputs can map to the same output. Understanding this definition is crucial for determining whether a given relation qualifies as a function.

Domain and Range

The domain of a relation consists of all possible input values, while the range includes all possible output values. Identifying the domain and range helps in understanding the behavior of the function and the set of values it can take. In the context of the given relation, analyzing the connections will reveal these sets.

Vertical Line Test

The vertical line test is a visual method used to determine if a relation is a function. If a vertical line intersects the graph of the relation at more than one point, the relation is not a function. This concept is particularly useful when analyzing graphical representations of relations, such as the one provided.