To find [tex]\( f(g(x)) \)[/tex] given that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex], let us first understand what it means for two functions to be inverses of each other.
Two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are considered inverse functions if:
[tex]\[ f(g(x)) = x \quad \text{for all } x \text{ in the domain of } g \][/tex]
[tex]\[ g(f(x)) = x \quad \text{for all } x \text{ in the domain of } f \][/tex]
This property defines the core relationship between a function and its inverse: they effectively undo each other's operations.
Therefore, if [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex], inputting [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex] will yield:
[tex]\[ f(g(x)) = x \][/tex]
This result holds true for any [tex]\( x \)[/tex]. The function [tex]\( f \)[/tex] composed with its inverse [tex]\( g \)[/tex] simply returns the input [tex]\( x \)[/tex] itself, exemplifying the defining characteristic of inverse functions.
Thus, the final solution to the problem is:
[tex]\[ f(g(x)) = x \][/tex]
This identity succinctly expresses the behavior expected when dealing with a function and its inverse, confirming that applying [tex]\( f \)[/tex] to [tex]\( g(x) \)[/tex] returns [tex]\( x \)[/tex].
