\{(-4,3),(-3,4),(0,3),(1,5),(2,6)\}

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01-11-2026
Mathematics

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\{(-4,3),(-3,4),(0,3),(1,5),(2,6)\}

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T20:36:55-05:00

A relation is a function if every input (x-coordinate) maps to exactly one output (y-coordinate).
For the set \(\{(-4, 3), (-3, 4), (0, 3), (1, 5), (2, 6)\}\), examine the domain values \(\{-4, -3, 0, 1, 2\}\).
Since each input value is unique, no input maps to multiple outputs.
The relation is therefore a function.
Final Answer: \(\boxed{True}\)

Explanation

Defining a Function and Evaluating the Relation

A relation is a set of ordered pairs that maps elements from a domain to a range. For a relation to be classified as a function, each element in its domain (the x-coordinate) must correspond to exactly one element in its range (the y-coordinate). In other words, if the same x-coordinate appears more than once in the set of ordered pairs, it must be paired with the same y-coordinate every time to maintain the functional relationship. If a single x-coordinate is paired with two or more different y-coordinates, the relation fails to be a function. Our set is given by \(\{(-4, 3), (-3, 4), (0, 3), (1, 5), (2, 6)\}\).

Inspecting the Input-Output Mapping

To determine if the relation is a function, we list the input values (the first coordinate of each pair) and their corresponding output values (the second coordinate of each pair):

\(-4\) maps to \(3\)
\(-3\) maps to \(4\)
\(0\) maps to \(3\)
\(1\) maps to \(5\)
\(2\) maps to \(6\)

Looking at this list, every input value in the domain \(\{-4, -3, 0, 1, 2\}\) is unique. Because no input value is repeated, there is no possibility for an input to map to more than one output. Therefore, every x-value has exactly one y-value associated with it.

Conclusion

Since every x-value in the domain corresponds to exactly one y-value, the given set of ordered pairs represents a valid function. The fact that the output value \(3\) is repeated (associated with both \(-4\) and \(0\)) does not violate the definition of a function; only the repetition of input values with different outputs would invalidate it. Thus, the relation is a function. The final answer is \(\boxed{True}\).

Examples

In daily life, functions are used to model predictable outcomes. For instance, consider a vending machine: if you press the button for a specific item (the input), you expect to receive that exact item (the output). If pressing the button labeled 'A1' sometimes resulted in a snack and other times resulted in a drink, the vending machine would not be functioning as a predictable 'function.' Similarly, in our set, each unique input leads to a specific output, making it a well-defined function.