If [tex]$f(x)=5 x-25$[/tex] and [tex]$g(x)=\frac{1}{5} x+5$[/tex], which expression could be used to verify [tex]$g(x)$[/tex] is the inverse of [tex]$f(x)$[/tex]?
A. [tex]$\frac{1}{5}(\frac{1}{5} x+5)+5$[/tex]
B. [tex]$\frac{1}{5}(5 x-25)+5$[/tex]
C. [tex]$\frac{1}{(\frac{1}{5} x+5)}$[/tex]
D. [tex]$5(\frac{1}{5} x+5)+5$[/tex]
Answer (1)
To verify if g ( x ) is the inverse of f ( x ) , we need to check if f ( g ( x )) = x or g ( f ( x )) = x .
Calculate g ( f ( x )) = 5 1 ( 5 x − 25 ) + 5 .
Simplify the expression: g ( f ( x )) = x − 5 + 5 = x .
The expression that verifies g ( x ) is the inverse of f ( x ) is 5 1 ( 5 x − 25 ) + 5 .
Explanation
Understanding Inverse Functions We are given two functions, f ( x ) = 5 x − 25 and g ( x ) = 5 1 x + 5 . We need to determine which expression can be used to verify if g ( x ) is the inverse of f ( x ) . A function g ( x ) is the inverse of f ( x ) if and only if f ( g ( x )) = x and g ( f ( x )) = x . Therefore, we need to check which of the given expressions equals x .
Checking Function Composition Let's evaluate f ( g ( x )) :
f ( g ( x )) = f ( 5 1 x + 5 ) = 5 ( 5 1 x + 5 ) − 25 = x + 25 − 25 = x Now let's evaluate g ( f ( x )) :
g ( f ( x )) = g ( 5 x − 25 ) = 5 1 ( 5 x − 25 ) + 5 = x − 5 + 5 = x Both compositions result in x , so both f ( g ( x )) and g ( f ( x )) can be used to verify that g ( x ) is the inverse of f ( x ) . However, we need to choose from the given options.
Analyzing the Options The given options are:
5 1 ( 5 1 x + 5 ) + 5
5 1 ( 5 x − 25 ) + 5
( 5 1 x + 5 ) 1
5 ( 5 1 x + 5 ) + 5
Option 1 represents 5 1 g ( x ) + 5 , which is not g ( f ( x )) .
Option 2 represents g ( f ( x )) = 5 1 ( 5 x − 25 ) + 5 , which we calculated to be x .
Option 3 represents g ( x ) 1 , which is not related to verifying the inverse. Option 4 represents f ( g ( x )) = 5 ( 5 1 x + 5 ) + 5 , but it should be 5 ( 5 1 x + 5 ) − 25 to equal x .
Final Answer Therefore, the correct expression to verify that g ( x ) is the inverse of f ( x ) is 5 1 ( 5 x − 25 ) + 5 , which simplifies to x .
Conclusion The expression that could be used to verify g ( x ) is the inverse of f ( x ) is 5 1 ( 5 x − 25 ) + 5 .
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 ) = 5 x − 25 is an encryption function, then g ( x ) = 5 1 x + 5 would be the decryption function. Applying f and then g (or vice versa) should return the original message.
