If $f(x)=5 x-25$ and $g(x)=\frac{1}{5} x+5$, which expression could be used to verify $g(x)$ is the inverse of $f(x)$?
A. $\frac{1}{5}(\frac{1}{5} x+5)+5$
B. $\frac{1}{5}(5 x-25)+5$
C. $\frac{1}{(\frac{1}{5} x+5)}$
D. $5(\frac{1}{5} x+5)+5$
Answer (2)
To verify if g ( x ) is the inverse of f ( x ) , we need to check if g ( f ( x )) = x or f ( g ( x )) = x .
Calculate g ( f ( x )) by substituting f ( x ) into g ( x ) : g ( f ( x )) = g ( 5 x − 25 ) = 5 1 ( 5 x − 25 ) + 5 .
The expression 5 1 ( 5 x − 25 ) + 5 can be used to verify if g ( x ) is the inverse of f ( x ) .
The correct expression is 5 1 ( 5 x − 25 ) + 5 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 5 x − 25 and g ( x ) = 5 1 x + 5 . We want to find an expression that verifies whether g ( x ) is the inverse of f ( x ) .
Inverse Function Verification To verify that g ( x ) is the inverse of f ( x ) , we need to check if f ( g ( x )) = x or g ( f ( x )) = x . This means we need to substitute g ( x ) into f ( x ) and see if we get x , or substitute f ( x ) into g ( x ) and see if we get x .
Calculating f(g(x)) Let's consider f ( g ( x )) . This means we substitute g ( x ) = 5 1 x + 5 into f ( x ) = 5 x − 25 . So we have: f ( g ( x )) = f ( 5 1 x + 5 ) = 5 ( 5 1 x + 5 ) − 25
Calculating g(f(x)) Now let's consider g ( f ( x )) . This means we substitute f ( x ) = 5 x − 25 into g ( x ) = 5 1 x + 5 . So we have: g ( f ( x )) = g ( 5 x − 25 ) = 5 1 ( 5 x − 25 ) + 5
Identifying the Correct Expression Comparing the expression for g ( f ( x )) with the given options, we see that the expression 5 1 ( 5 x − 25 ) + 5 matches one of the options.
Final Answer Therefore, 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 encoding and decoding messages. If f ( x ) is an encoding function, then its inverse g ( x ) is the decoding function. Verifying that g ( x ) is indeed the inverse of f ( x ) ensures that the encoded message can be correctly decoded back to its original form. For example, if f ( x ) = 5 x − 25 encodes a message, then g ( x ) = 5 1 x + 5 decodes it. We verify this by checking if g ( f ( x )) = x , ensuring the original message is recovered.
To verify that g ( x ) is the inverse of f ( x ) , we calculate g ( f ( x )) and find it equals x . The expression 5 1 ( 5 x − 25 ) + 5 confirms this relationship. Thus, the chosen option is B.
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