Answer :
To determine the value of [tex]\( h(f(x)) \)[/tex] when [tex]\( h(x) \)[/tex] is the inverse function of [tex]\( f(x) \)[/tex], we need to understand the relationship between a function and its inverse.
By definition, if [tex]\( h(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex], then:
1. [tex]\( f(h(x)) = x \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( h \)[/tex].
2. [tex]\( h(f(x)) = x \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( f \)[/tex].
This relationship means that applying the function [tex]\( f \)[/tex] and then applying its inverse [tex]\( h \)[/tex] should return us to the original input [tex]\( x \)[/tex].
To put it another way, the inverse function [tex]\( h(x) \)[/tex] reverses the operation of [tex]\( f(x) \)[/tex]. Thus, when you first apply [tex]\( f(x) \)[/tex] to some value [tex]\( x \)[/tex], and then apply [tex]\( h(x) \)[/tex] to the result of [tex]\( f(x) \)[/tex], you should obtain the original value [tex]\( x \)[/tex] with which you started.
Therefore, the value of [tex]\( h(f(x)) \)[/tex] is [tex]\( x \)[/tex].
So the answer is:
[tex]\[ \boxed{x} \][/tex]
By definition, if [tex]\( h(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex], then:
1. [tex]\( f(h(x)) = x \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( h \)[/tex].
2. [tex]\( h(f(x)) = x \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( f \)[/tex].
This relationship means that applying the function [tex]\( f \)[/tex] and then applying its inverse [tex]\( h \)[/tex] should return us to the original input [tex]\( x \)[/tex].
To put it another way, the inverse function [tex]\( h(x) \)[/tex] reverses the operation of [tex]\( f(x) \)[/tex]. Thus, when you first apply [tex]\( f(x) \)[/tex] to some value [tex]\( x \)[/tex], and then apply [tex]\( h(x) \)[/tex] to the result of [tex]\( f(x) \)[/tex], you should obtain the original value [tex]\( x \)[/tex] with which you started.
Therefore, the value of [tex]\( h(f(x)) \)[/tex] is [tex]\( x \)[/tex].
So the answer is:
[tex]\[ \boxed{x} \][/tex]
