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]
