Which function has an inverse that is also a function?
$\lbrace(-4,3),(-2,7),(-1,0),(4,-3),(11,-7)\rbrace$
$\lbrace(-4,6),(-2,2),(-1,6),(4,2),(11,2)\rbrace$
$\lbrace(-4,5),(-2,9),(-1,8),(4,8),(11,4)\rbrace$
$\lbrace(-4,4),(-2,-1),(-1,0),(4,1),(11,1)\rbrace$
Answer (1)
A function has an inverse that is also a function if it is one-to-one, meaning each y -value corresponds to a unique x -value. The first set {( − 4 , 3 ) , ( − 2 , 7 ) , ( − 1 , 0 ) , ( 4 , − 3 ) , ( 11 , − 7 )} has unique y -values, so it has an inverse that is also a function. {( − 4 , 3 ) , ( − 2 , 7 ) , ( − 1 , 0 ) , ( 4 , − 3 ) , ( 11 , − 7 )}
Explanation
Understanding the Problem A function has an inverse that is also a function if and only if the original function is one-to-one. A function is one-to-one if each y -value corresponds to a unique x -value. We need to check each set of ordered pairs to see if any y -values are repeated.
Analyzing the First Set Let's examine the first set: {( − 4 , 3 ) , ( − 2 , 7 ) , ( − 1 , 0 ) , ( 4 , − 3 ) , ( 11 , − 7 )} . The y -values are 3 , 7 , 0 , − 3 , − 7 . Since all y -values are unique, this function has an inverse that is also a function.
Analyzing the Second Set Let's examine the second set: {( − 4 , 6 ) , ( − 2 , 2 ) , ( − 1 , 6 ) , ( 4 , 2 ) , ( 11 , 2 )} . The y -values are 6 , 2 , 6 , 2 , 2 . Since the y -values 6 and 2 are repeated, this function does not have an inverse that is also a function.
Analyzing the Third Set Let's examine the third set: {( − 4 , 5 ) , ( − 2 , 9 ) , ( − 1 , 8 ) , ( 4 , 8 ) , ( 11 , 4 )} . The y -values are 5 , 9 , 8 , 8 , 4 . Since the y -value 8 is repeated, this function does not have an inverse that is also a function.
Analyzing the Fourth Set Let's examine the fourth set: {( − 4 , 4 ) , ( − 2 , − 1 ) , ( − 1 , 0 ) , ( 4 , 1 ) , ( 11 , 1 )} . The y -values are 4 , − 1 , 0 , 1 , 1 . Since the y -value 1 is repeated, this function does not have an inverse that is also a function.
Conclusion Therefore, the first set {( − 4 , 3 ) , ( − 2 , 7 ) , ( − 1 , 0 ) , ( 4 , − 3 ) , ( 11 , − 7 )} is the only function that has an inverse that 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 reverse the encryption process and access sensitive information. Therefore, ensuring that the encryption function has a unique output for each input is essential for maintaining data security.
