Find $f(g(x))$ and $g(f(x))$ and determine whether the pair of functions $f$ and $g$ are inverses of each other.
$f(x)=5 x-2 \text { and } g(x)=\frac{x+5}{2}$
a. $f(g(x))=$ $\square$
(Simplify your answer. Use integers or fractions for any numbers in the expression.)
b. $g(f(x))=$ $\square$
(Simplify your answer. Use integers or fractions for any numbers in the expression.)
c.
$f$ and $g$ are not inverses of each other.
$f$ and $g$ are inverses of each other.
Answer (2)
Calculate f ( g ( x )) by substituting g ( x ) into f ( x ) and simplifying: f ( g ( x )) = 2 5 x + 21 .
Calculate g ( f ( x )) by substituting f ( x ) into g ( x ) and simplifying: g ( f ( x )) = 2 5 x + 3 .
Check if f ( g ( x )) = x and g ( f ( x )) = x . Since this is not the case, the functions are not inverses.
Conclude that f ( g ( x )) = 2 5 x + 21 , g ( f ( x )) = 2 5 x + 3 , and the functions are not inverses of each other. f and g are not inverses of each other.
Explanation
Understanding the Problem We are given two functions, f ( x ) = 5 x − 2 and g ( x ) = 2 x + 5 . We need to find f ( g ( x )) and g ( f ( x )) and then determine if these two functions are inverses of each other. Two functions are inverses if and only if f ( g ( x )) = x and g ( f ( x )) = x .
Calculating f(g(x)) First, let's find f ( g ( x )) . We substitute g ( x ) into f ( x ) : f ( g ( x )) = f ( 2 x + 5 ) = 5 ( 2 x + 5 ) − 2
Now, we simplify the expression: f ( g ( x )) = 2 5 ( x + 5 ) − 2 = 2 5 x + 25 − 2 = 2 5 x + 25 − 2 4 = 2 5 x + 21
Calculating g(f(x)) Next, let's find g ( f ( x )) . We substitute f ( x ) into g ( x ) : g ( f ( x )) = g ( 5 x − 2 ) = 2 ( 5 x − 2 ) + 5
Now, we simplify the expression: g ( f ( x )) = 2 5 x − 2 + 5 = 2 5 x + 3
Checking for Inverses Now we check if f ( g ( x )) = x and g ( f ( x )) = x . We found that f ( g ( x )) = 2 5 x + 21 and g ( f ( x )) = 2 5 x + 3 . Since 2 5 x + 21 = x and 2 5 x + 3 = x , the functions f ( x ) and g ( x ) are not inverses of each other.
Final Answer Therefore, f ( g ( x )) = 2 5 x + 21 and g ( f ( x )) = 2 5 x + 3 , and f and g are not inverses of each other.
Examples
In cryptography, inverse functions are crucial for encoding and decoding messages. If f ( x ) represents an encoding function, its inverse g ( x ) would be the decoding function. If f ( x ) and g ( x ) were inverses, applying f and then g would return the original message, ensuring secure communication. However, if they are not inverses, the decoding process would not correctly recover the original message, compromising security.
The compositions of the functions yield f ( g ( x )) = 2 5 x + 21 and g ( f ( x )) = 2 5 x + 3 . Since neither composition equals x , the functions f and g are not inverses of each other. Therefore, the final conclusion is that f and g are not inverses of each other.
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