Determine whether each relation defines a function. See Example 1. x y 3 -4 7 -410 -4

Verified step by step guidance

1. Identify the given relation as a set of ordered pairs: \((3, -4), (7, -4), (10, -4)\).
2. Recall the definition of a function: A relation is a function if each input (x-value) is associated with exactly one output (y-value).
3. Examine each ordered pair to determine if any x-value is repeated with a different y-value.
4. Observe that each x-value (3, 7, 10) is unique and is paired with the same y-value (-4).
5. Conclude that since no x-value is repeated with a different y-value, the relation defines a function.

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 for every unique value of 'x', there cannot be multiple corresponding 'y' values. Understanding this definition is crucial for determining if a given relation qualifies as a function.

Mapping of Inputs to Outputs

In the context of functions, mapping refers to how each input is paired with an output. For the relation to be a function, each input must map to a single output. In the provided relation, we analyze the 'x' values to see if any repeat with different 'y' values, which would violate the function rule.

Vertical Line Test

The vertical line test is a visual method used to determine if a relation is a function. If a vertical line intersects the graph of the relation at more than one point, the relation is not a function. This concept helps in visualizing the relationship between inputs and outputs, reinforcing the definition of a function.