Determine whether each relation defines a function. See Example 1.

Verified step by step guidance

1. Step 1: Understand the definition of a function. A relation is a function if each input (or x-value) is associated with exactly one output (or y-value).
2. Step 2: Identify the set of ordered pairs in the given relation. An ordered pair is typically in the form (x, y).
3. Step 3: Examine the x-values (inputs) in the relation. Check if any x-value is repeated with a different y-value.
4. Step 4: If no x-value is repeated with different y-values, the relation is a function. If any x-value is associated with more than one y-value, the relation is not a function.
5. Step 5: Conclude whether the relation is a function based on your examination of the x-values and their corresponding y-values.

Key Concepts

Relation

A relation is a set of ordered pairs, where each pair consists of an input (or domain element) and an output (or range element). In the context of functions, a relation can be represented as a graph, a table, or a set of points. Understanding relations is crucial for determining whether a specific relation meets the criteria to be classified as a function.

Function

A function is a specific type of relation where each input is associated with exactly one output. This means that for every x-value in the domain, there is only one corresponding y-value in the range. To determine if a relation is a function, one can use the vertical line test on its graph: if a vertical line intersects the graph at more than one point, the relation is not a function.

Vertical Line Test

The vertical line test is a visual method used to determine if a relation is a function by examining its graph. If any vertical line drawn through the graph intersects it at more than one point, the relation fails the test and is not a function. This test provides a straightforward way to assess the uniqueness of outputs for each input in a relation.