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

Verified step by step guidance

1. Identify the definition of a function: A relation is a function if each input (x-value) is associated with exactly one output (y-value).
2. List the x-values from the given set of ordered pairs: {5, 3, 4, 7}.
3. Check if any x-value is repeated in the list. If an x-value appears more than once with different y-values, the relation is not a function.
4. Since all x-values are unique in this set, each input is associated with exactly one output.
5. Conclude that the given relation defines a function because no x-value is repeated with different y-values.

Key Concepts

Definition of a Function

A function is a relation where each input (or 'x' value) is associated with exactly one output (or 'y' value). This means that no two ordered pairs can have the same first element with different second elements. For example, in the relation {(5, 1), (3, 2)}, both 5 and 3 are unique inputs, making it 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. The order is crucial because (x1, y1) is different from (y1, x1). In the given relation, each pair represents a mapping from an input to an output, which is essential for determining if the relation is 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 helps reinforce the idea that each input must correspond to a single output, which is fundamental in analyzing relations.