Which function has an inverse that is also a function?
{(-1,-2),(0,4),(1,3),(5,14),(7,4)}
{(-1,2),(0,4),(1,5),(5,4),(7,2)}
{(-1,3),(0,4),(1,14),(5,6),(7,2)}
{(-1,4),(0,4),(1,2),(5,3),(7,1)}
Answer (2)
A function has an inverse that is also a function if it is one-to-one.
A one-to-one function has unique y-values for each x-value.
Check each relation for repeated y-values.
The relation {( − 1 , 3 ) , ( 0 , 4 ) , ( 1 , 14 ) , ( 5 , 6 ) , ( 7 , 2 )} has no repeated y-values, so its inverse is also a function. {( − 1 , 3 ) , ( 0 , 4 ) , ( 1 , 14 ) , ( 5 , 6 ) , ( 7 , 2 )} .
Explanation
Understanding the Problem We are given four sets of ordered pairs, each representing a relation. We need to determine which relation has an inverse that is also a function. A function has an inverse that is also a function if and only if it is a one-to-one function (i.e., it passes the horizontal line test). A one-to-one function has the property that each x-value corresponds to a unique y-value, and each y-value corresponds to a unique x-value.
Condition for Inverse Function For a function to have an inverse that is also a function, no two distinct x-values can map to the same y-value. In other words, the function must be one-to-one. We can determine which of the given relations has an inverse that is also a function by checking if any y-values are repeated in each relation.
Checking Each Relation Let's examine each relation:
{( − 1 , − 2 ) , ( 0 , 4 ) , ( 1 , 3 ) , ( 5 , 14 ) , ( 7 , 4 )} : The y-value 4 is repeated (0,4) and (7,4). Therefore, the inverse is not a function.
{( − 1 , 2 ) , ( 0 , 4 ) , ( 1 , 5 ) , ( 5 , 4 ) , ( 7 , 2 )} : The y-values 2 and 4 are repeated. Therefore, the inverse is not a function.
{( − 1 , 3 ) , ( 0 , 4 ) , ( 1 , 14 ) , ( 5 , 6 ) , ( 7 , 2 )} : No y-values are repeated. Therefore, the inverse is a function.
{( − 1 , 4 ) , ( 0 , 4 ) , ( 1 , 2 ) , ( 5 , 3 ) , ( 7 , 1 )} : The y-value 4 is repeated. Therefore, the inverse is not a function.
Final Answer The relation with no repeated y-values is {( − 1 , 3 ) , ( 0 , 4 ) , ( 1 , 14 ) , ( 5 , 6 ) , ( 7 , 2 )} . Therefore, this is the function whose inverse is also a function.
Examples
In cryptography, one-to-one functions are crucial for creating secure encryption keys. If a function used for encryption is not one-to-one, it becomes easier for unauthorized parties to decrypt the message, compromising its security. Similarly, in data compression, one-to-one functions ensure that no information is lost when compressing and decompressing data, maintaining the integrity of the original data.
The relation {(-1,3),(0,4),(1,14),(5,6),(7,2)} is the only one with unique y-values for each x-value, meaning its inverse will also be a function. Therefore, the answer is this relation.
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