Determine whether each relation defines a function. See Example 1. {(2,4),(0,2),(2,6)}

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 and outputs in the given relation. The relation is given as a set of ordered pairs: \((2,4), (0,2), (2,6)\).
3. Step 3: List the x-values (inputs) from the ordered pairs. Here, the x-values are 2, 0, and 2.
4. Step 4: Check if any x-value is repeated with different y-values. In this relation, the x-value 2 is associated with both 4 and 6.
5. Step 5: Conclude whether the relation is a function. Since the x-value 2 is paired with two different y-values, the relation does not define a function.

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 if a 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 given, the pairs are (2,4), (0,2), and (2,6). Analyzing these pairs helps identify 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. This concept is useful for visual learners and can be applied to the set of ordered pairs to confirm the function's validity.