Determine whether each relation defines a function. See Example 1. {(5,1),(3,2),(4,9),(7,8)}

Verified step by step guidance

1. Understand the definition of a function: A relation is a function if each input (or domain value) is associated with exactly one output (or range value).
2. Identify the inputs (domain) from the given relation: The inputs are the first elements of each ordered pair, which are 5, 3, 4, and 7.
3. Check for any repeated inputs: Look at the list of inputs to see if any input value appears more than once.
4. Verify the uniqueness of outputs for each input: Since no input is repeated, each input is associated with only one output.
5. Conclude whether the relation is a function: Since each input has a unique output, the relation defines a function.

Key Concepts

Definition of a Function

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 for every x-value in the set of ordered pairs, there must be a unique y-value. If any x-value corresponds to more than one y-value, the relation cannot be classified as a function.

Ordered Pairs

Ordered pairs are pairs of elements written in the form (x, y), where x is the first element and y is the second. In the context of relations, these pairs represent the relationship between two sets, typically the domain and range. Analyzing ordered pairs helps determine if a relation meets the criteria to be a function by checking the uniqueness of y-values for each x-value.

Vertical Line Test

The vertical line test is a visual method used to determine if a graph represents a function. If any vertical line drawn on the graph intersects the curve at more than one point, the relation is not a function. This test reinforces the concept that each input must correspond to a single output, providing a practical way to assess functions graphically.