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Answer :
Sure, let's go through the details of each part of the question regarding functions and types of relations:
a. One-to-Many Relations:
- If you have a relation where one x-value maps to multiple y-values, this is called a "one-to-many" relation.
- Such a relation cannot be considered a function. This is because functions require that each input (x-value) corresponds to exactly one output (y-value).
- If you reverse the x- and y-values in a one-to-many relation, you check if each y-value now corresponds to exactly one x-value. If this condition is met, the reversed relation is a function because each y-value maps to one unique x-value.
b. Not One-to-One Relations:
- A relation is not one-to-one if there is at least one y-value mapping to multiple x-values. However, it can still be considered a function as long as the original condition of each x-value mapping to a single y-value is met.
- When you reverse x- and y-values in a relation that is not one-to-one, it becomes important to check if the reversed relation is one-to-many. If in the reversed relation each former y-value (now x-value) maps to exactly one former x-value (now y-value), then the reversed relation is a function.
c. One-to-One Relations:
- A one-to-one relation is a special type of function where each x-value maps to exactly one unique y-value, and each y-value also maps to exactly one unique x-value.
- This kind of relation is always a function because it satisfies the condition for the definition of a function.
- If you reverse the x- and y-values in a one-to-one relation, the reversed relation also remains a function. This is because the one-to-one nature of the relationship means that the reversal causes no loss in the unique mapping necessary for it to still be a function.
By understanding and applying these principles, you can determine whether a given relation is a function and how it behaves if you switch its input and output values.
a. One-to-Many Relations:
- If you have a relation where one x-value maps to multiple y-values, this is called a "one-to-many" relation.
- Such a relation cannot be considered a function. This is because functions require that each input (x-value) corresponds to exactly one output (y-value).
- If you reverse the x- and y-values in a one-to-many relation, you check if each y-value now corresponds to exactly one x-value. If this condition is met, the reversed relation is a function because each y-value maps to one unique x-value.
b. Not One-to-One Relations:
- A relation is not one-to-one if there is at least one y-value mapping to multiple x-values. However, it can still be considered a function as long as the original condition of each x-value mapping to a single y-value is met.
- When you reverse x- and y-values in a relation that is not one-to-one, it becomes important to check if the reversed relation is one-to-many. If in the reversed relation each former y-value (now x-value) maps to exactly one former x-value (now y-value), then the reversed relation is a function.
c. One-to-One Relations:
- A one-to-one relation is a special type of function where each x-value maps to exactly one unique y-value, and each y-value also maps to exactly one unique x-value.
- This kind of relation is always a function because it satisfies the condition for the definition of a function.
- If you reverse the x- and y-values in a one-to-one relation, the reversed relation also remains a function. This is because the one-to-one nature of the relationship means that the reversal causes no loss in the unique mapping necessary for it to still be a function.
By understanding and applying these principles, you can determine whether a given relation is a function and how it behaves if you switch its input and output values.
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