- A relation is a function only if each input \(x\) maps to exactly one output \(y\).
- In the first relation, input \(2\) maps to both \(-9\) and \(8\), so it is not a function.
- In the second relation, inputs \(-8\) and \(1\) map to multiple outputs, so it is not a function.
- In the third relation, input \(-5\) maps to four different outputs, so it is not a function.
- Therefore, \(\boxed{\text{None of the relations are functions}}\).
Explanation
- Understanding the Definition of a Function
A relation is defined as a function if every input value \(x\) in the domain corresponds to exactly one unique output value \(y\) in the codomain. To determine if a relation is a function, we must check if any input value \(x\) is associated with more than one output value \(y\). If we find an \(x\) value that has multiple \(y\) values, that relation is not a function.
- Analyzing the First Relation
Let's analyze the first table:
- Input \(-1\) maps to \(7\).
- Input \(2\) maps to \(-9\).
- Input \(2\) also maps to \(8\).
Because the input \(2\) is associated with two different output values (\(-9\) and \(8\)), the first relation does not satisfy the definition of a function.
- Analyzing the Second Relation
Let's analyze the second table:
- Input \(-8\) maps to \(-9\).
- Input \(-8\) also maps to \(2\).
- Input \(1\) maps to \(-9\).
- Input \(1\) also maps to \(2\).
Because the input \(-8\) is associated with two different output values (\(-9\) and \(2\)), and the input \(1\) is also associated with two different output values (\(-9\) and \(2\)), the second relation is not a function.
- Analyzing the Third Relation
Let's analyze the third table:
- Input \(-5\) maps to \(1\), \(7\), \(-9\), and \(2\).
Since a single input \(-5\) maps to four different output values, the third relation is not a function.
- Conclusion
Based on the analysis, every relation provided contains at least one input value that maps to multiple different output values. Therefore, none of the relations are functions of \(x\). The final answer is that none of the listed relations are functions.
Examples
Understanding functions is crucial in real-world scenarios like data processing or programming. For example, if you are designing a database for a library, each unique Book ID (the input \(x\)) should point to only one specific book title (the output \(y\)). If one Book ID returned multiple titles, your database would be unreliable, much like the relations analyzed here that fail to be functions because they map a single input to multiple outputs. Ensuring that relations are functions helps maintain order and predictability in systems.