Which statement verifies that [tex]f(x)[/tex] and [tex]g(x)[/tex] are inverses of each other?
A. [tex]f(g(x))=x[/tex]
B. [tex]f(g(x))=x[/tex] and [tex]g(f(x))=-x[/tex]
C. [tex]f(g(x))=\frac{1}{g(f(x))}[/tex]
D. [tex]f(g(x))=x[/tex] and [tex]g(f(x))=x[/tex]
Answer (1)
To verify if two functions f ( x ) and g ( x ) are inverses, we need to check if f ( g ( x )) = x and g ( f ( x )) = x .
Option 1 only provides f ( g ( x )) = x , which is insufficient.
Option 2 states f ( g ( x )) = x and g ( f ( x )) = − x , which is incorrect because it should be g ( f ( x )) = x .
Option 3 gives f ( g ( x )) = g ( f ( x )) 1 , which is also incorrect.
Option 4 provides the correct condition: f ( g ( x )) = x and g ( f ( x )) = x .
Therefore, the correct answer is: f ( g ( x )) = x and g ( f ( x )) = x .
Explanation
Understanding Inverse Functions To verify that two functions, f ( x ) and g ( x ) , are inverses of each other, we need to check the composition of the functions in both orders. The composition of a function and its inverse should result in the identity function, which is simply x .
Analyzing the Options The condition for f ( x ) and g ( x ) to be inverses is that f ( g ( x )) = x and g ( f ( x )) = x . Let's examine each option to see which one satisfies this condition.
Evaluating Option 1 Option 1: f ( g ( x )) = x . This is only half of the requirement. We also need to check g ( f ( x )) = x . So, this option is not sufficient.
Evaluating Option 2 Option 2: f ( g ( x )) = x and g ( f ( x )) = − x . This is incorrect because g ( f ( x )) should be equal to x , not − x .
Evaluating Option 3 Option 3: f ( g ( x )) = g ( f ( x )) 1 . This is incorrect. For inverse functions, f ( g ( x )) = x and g ( f ( x )) = x , so we should have x = x 1 , which is not true for all x .
Evaluating Option 4 Option 4: f ( g ( x )) = x and g ( f ( x )) = x . This is the correct condition for inverse functions. Therefore, this is the statement that verifies that f ( x ) and g ( x ) are inverses of each other.
Final Answer The correct statement that verifies that f ( x ) and g ( x ) are inverses of each other is f ( g ( x )) = x and g ( f ( x )) = x .
Examples
In cryptography, inverse functions are used for encryption and decryption. If f ( x ) encrypts a message x , then its inverse function g ( x ) decrypts the encrypted message back to the original message. For example, if f ( x ) = x + 5 (mod 26) is used to encrypt a letter, then g ( x ) = x − 5 (mod 26) would be used to decrypt it. The property f ( g ( x )) = x ensures that the decryption process correctly recovers the original message.
