If [tex]$f(x)=6 x-1$[/tex] and [tex]$g(x)=\frac{x+1}{6}$[/tex], which expressions can be used to verify [tex]$g(x)$[/tex] is the inverse of [tex]$f(x)$[/tex]? Check all that apply.
[tex]$6(6 x-1)-1$[/tex]
[tex]$\frac{(6 x-1)+1}{6}$[/tex]
[tex]$6\left(\frac{x+1}{6}\right)-1$[/tex]
[tex]$\frac{\left(\frac{x+1}{6}\right)+1}{6}$[/tex]
[tex]$\frac{x+1}{6}+6 x-1$[/tex]
Answer (1)
To verify if g ( x ) is the inverse of f ( x ) , we need to check if f ( g ( x )) = x and g ( f ( x )) = x .
Evaluate f ( g ( x )) = 6 ( 6 x + 1 ) − 1 .
Evaluate g ( f ( x )) = 6 ( 6 x − 1 ) + 1 .
The expressions that can be used to verify that g ( x ) is the inverse of f ( x ) are 6 ( 6 x − 1 ) + 1 and 6 ( 6 x + 1 ) − 1 .
6 ( 6 x − 1 ) + 1 , 6 ( 6 x + 1 ) − 1
Explanation
Understanding Inverse Functions We are given two functions, f ( x ) = 6 x − 1 and g ( x ) = 6 x + 1 , and we want to determine which expressions can be used to verify if g ( x ) is the inverse of f ( x ) . To verify that g ( x ) is the inverse of f ( x ) , we need to check if f ( g ( x )) = x and g ( f ( x )) = x .
Evaluating f(g(x)) Let's evaluate f ( g ( x )) . This means we substitute g ( x ) into f ( x ) : f ( g ( x )) = 6 ( g ( x )) − 1 = 6 ( 6 x + 1 ) − 1
Evaluating g(f(x)) Now let's evaluate g ( f ( x )) . This means we substitute f ( x ) into g ( x ) : g ( f ( x )) = 6 f ( x ) + 1 = 6 ( 6 x − 1 ) + 1
Comparing with Given Expressions Now we compare the given expressions with f ( g ( x )) and g ( f ( x )) to see which ones match.
6 ( 6 x − 1 ) − 1 : This is f ( f ( x )) , not used for verifying inverse functions.
6 ( 6 x − 1 ) + 1 : This is g ( f ( x )) , and it should simplify to x if g ( x ) is the inverse of f ( x ) .
6 ( 6 x + 1 ) − 1 : This is f ( g ( x )) , and it should simplify to x if g ( x ) is the inverse of f ( x ) .
6 ( 6 x + 1 ) + 1 : This is g ( g ( x )) , not used for verifying inverse functions.
6 x + 1 + 6 x − 1 : This is g ( x ) + f ( x ) , not used for verifying inverse functions.
Final Answer Therefore, the expressions that can be used to verify that g ( x ) is the inverse of f ( x ) are:
6 ( 6 x − 1 ) + 1 and 6 ( 6 x + 1 ) − 1
Examples
In cryptography, inverse functions are used for encryption and decryption. If f ( x ) encrypts a message x , then its inverse g ( x ) decrypts the encrypted message back to the original message. For example, if f ( x ) = 6 x − 1 is an encryption function, then g ( x ) = 6 x + 1 would be the decryption function. Verifying that g ( x ) is indeed the inverse of f ( x ) ensures that the decryption process works correctly, allowing secure communication.
