If [tex][tex]$h(x)$[/tex][/tex] is the inverse of [tex][tex]$f(x)$[/tex][/tex], what is the value of [tex][tex]$h(f(x))$[/tex][/tex]?

A. 0
B. 1
C. [tex][tex]$x$[/tex][/tex]
D. [tex][tex]$f(x)$[/tex][/tex]



Answer :

To understand what happens when you apply a function and then its inverse, we need to review some properties of inverse functions.

Given a function [tex]\( f(x) \)[/tex] and its inverse [tex]\( h(x) \)[/tex], by definition, these two functions undo each other's effects. This is fundamental to the concept of inverse functions. Thus, if you first apply [tex]\( f \)[/tex] and then [tex]\( h \)[/tex] to an input [tex]\( x \)[/tex], you should get back your original input [tex]\( x \)[/tex].

Formally, if [tex]\( h(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex], then:
[tex]\[ h(f(x)) = x \][/tex]

This statement means that when you apply [tex]\( f \)[/tex] to [tex]\( x \)[/tex] and then apply [tex]\( h \)[/tex] to the result [tex]\( f(x) \)[/tex], you retrieve the initial value [tex]\( x \)[/tex].

Let's break it down step-by-step:
1. You start with an input [tex]\( x \)[/tex].
2. Apply the function [tex]\( f \)[/tex] to this input to get [tex]\( f(x) \)[/tex].
3. Then, apply the inverse function [tex]\( h \)[/tex] to the result [tex]\( f(x) \)[/tex].

Because [tex]\( h(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex], applying [tex]\( h \)[/tex] after [tex]\( f \)[/tex] should return you to the original input value [tex]\( x \)[/tex].

Therefore, the value of [tex]\( h(f(x)) \)[/tex] is:
[tex]\[ x \][/tex]