Determine whether each relation defines a function. See Example 1. {(-3,1),(4,1),(-2,7)}

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 inputs (x-values) in the given relation. The inputs are -3, 4, and -2.
3. Step 3: Check if any input is repeated with a different output. In this relation, each input appears only once: -3 is paired with 1, 4 is paired with 1, and -2 is paired with 7.
4. Step 4: Since no input is associated with more than one output, the relation satisfies the condition of a function.
5. Step 5: Conclude that the given relation is a function because each input has a unique output.

Key Concepts

Definition of a Function

A function is a specific type of relation where each input (or 'x' value) is associated with exactly one output (or 'y' value). This means that for any given x, there cannot be two different y values. Understanding this definition is crucial for determining whether a given relation qualifies as a function.

Ordered Pairs

Relations are often represented as sets of ordered pairs, where each pair consists of an input and its corresponding output. In the example provided, the pairs are {(-3,1), (4,1), (-2,7)}. Analyzing these pairs helps in identifying if any input is repeated with different outputs, which would disqualify the relation from being a function.

Vertical Line Test

The vertical line test is a visual method used to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the relation is not a function. While this test applies to graphical representations, it reinforces the concept that each input must map to a single output.