Verified Answer :

  • Define the domain as the set of unique x-values and the range as the set of unique y-values.
  • Determine if a relation is a function by checking if every x-value maps to exactly one y-value.
  • In Relation 1 \(\{(-6,4), (5,-1), (0,3), (-2,4)\}\), all x-values are unique, so it is a function.
  • In Relation 2, x-value \(-3\) maps to both \(1\) and \(2\), so it is not a function.

Explanation

  1. Analysis of Relations and Functions

To determine whether a relation is a function, we must first analyze the domain and the range of the relation. The domain consists of all unique x-coordinates (input values), and the range consists of all unique y-coordinates (output values). A relation is considered a function if and only if every input value in the domain is paired with exactly one output value in the range. If any x-coordinate corresponds to more than one y-coordinate, the relation fails to be a function. We evaluate this by checking if any x-value appears more than once in the set of ordered pairs. If it does, we must ensure that the associated y-values are identical.

  1. Evaluating Relation 1

For the first relation \(\{(-6,4), (5,-1), (0,3), (-2,4)\}\):

  • The x-coordinates are \(-6, 5, 0, -2\). Thus, the domain is \(\{-6, -2, 0, 5\}\).
  • The y-coordinates are \(4, -1, 3, 4\). The unique y-values are \(\{-1, 3, 4\}\). Thus, the range is \(\{-1, 3, 4\}\).
  • Since each x-coordinate appears only once and maps to a unique y-value, this relation is a function.
  1. Evaluating Relation 2

For the second relation, represented by the table:
| x | y |
| :-: | :-: |
| -4 | 8 |
| -3 | 1 |
| 0 | -5 |
| -3 | 2 |

  • The unique x-coordinates are \(-4, -3, 0\). Thus, the domain is \(\{-4, -3, 0\}\).
  • The y-coordinates are \(8, 1, -5, 2\). Thus, the range is \(\{-5, 1, 2, 8\}\).
  • Notice that the x-coordinate \(-3\) appears twice. It maps to \(y = 1\) and \(y = 2\). Because one input maps to two different outputs, this relation is not a function.

Examples

In a real-world scenario, you can think of a function as a vending machine. If you press a specific button (the input or x-value), you expect one specific snack (the output or y-value) to come out every time. If you press button A and sometimes a bag of chips comes out, but other times a candy bar comes out, the machine is not functioning correctly—it's not a 'function'. Similarly, in mathematics, each input must reliably produce exactly one output. This is essential in fields like computer programming, where a command must consistently trigger the same result to be reliable.

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