If [tex]f(x)=x+3[/tex] and [tex]g(x)=x-3[/tex],
(a) [tex]f(g(x))=[/tex] $\square$
(b) [tex]g(f(x))=[/tex] $\square$
(c) Thus [tex]g(x)[/tex] is called an $\square$ function of [tex]f(x)[/tex]
Answer (2)
Find f ( g ( x )) by substituting g ( x ) into f ( x ) : f ( g ( x )) = f ( x − 3 ) = ( x − 3 ) + 3 = x .
Find g ( f ( x )) by substituting f ( x ) into g ( x ) : g ( f ( x )) = g ( x + 3 ) = ( x + 3 ) − 3 = x .
Since f ( g ( x )) = x and g ( f ( x )) = x , conclude that g ( x ) is the inverse function of f ( x ) .
The final answer is: f ( g ( x )) = x , g ( f ( x )) = x , and g ( x ) is the inverse function of f ( x ) , so the answers are x , x , and in v erse .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x + 3 and g ( x ) = x − 3 . We need to find f ( g ( x )) , g ( f ( x )) , and determine the relationship between f ( x ) and g ( x ) .
Finding f(g(x)) To find f ( g ( x )) , we substitute g ( x ) into f ( x ) . So, f ( g ( x )) = f ( x − 3 ) = ( x − 3 ) + 3 .
Simplifying f(g(x)) Simplifying the expression, we get f ( g ( x )) = x − 3 + 3 = x .
Finding g(f(x)) To find g ( f ( x )) , we substitute f ( x ) into g ( x ) . So, g ( f ( x )) = g ( x + 3 ) = ( x + 3 ) − 3 .
Simplifying g(f(x)) Simplifying the expression, we get g ( f ( x )) = x + 3 − 3 = x .
Determining the Relationship Since f ( g ( x )) = x and g ( f ( x )) = x , g ( x ) is the inverse function of f ( x ) .
Examples
In cryptography, inverse functions can be used for encoding and decoding messages. If f ( x ) is an encoding function, then g ( x ) would be the decoding function, allowing you to retrieve the original message. For example, if f ( x ) = x + 3 is used to encode a letter by shifting it three positions forward, then g ( x ) = x − 3 would decode the message by shifting it three positions backward. This concept ensures secure communication by easily converting messages back and forth.
We found that f(g(x)) = x and g(f(x)) = x, which shows that g(x) is the inverse function of f(x). The relationships demonstrate that each function reverses the other. Thus, the answers are: f(g(x)) = x, g(f(x)) = x, and g(x) is the inverse function.
;
