To determine the value of [tex]\( h(f(x)) \)[/tex] given that [tex]\( h(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex], let's go through the concepts and definitions step by step:
1. Definition of Inverse Functions:
- If [tex]\( h(x) \)[/tex] is the inverse function of [tex]\( f(x) \)[/tex], it means that applying [tex]\( h \)[/tex] to [tex]\( f \)[/tex] will yield the original input value. Mathematically, this is expressed as:
[tex]\[
h(f(x)) = x
\][/tex]
2. Understanding the Identity:
- By definition, for two functions [tex]\( f \)[/tex] and [tex]\( h \)[/tex] to be inverses of each other, we must have:
[tex]\[
h(f(x)) = x \text{ and } f(h(x)) = x
\][/tex]
- This means that applying [tex]\( f \)[/tex] and then [tex]\( h \)[/tex] (or vice-versa) should return the input value [tex]\( x \)[/tex].
3. Application to the Question:
- When we have [tex]\( h(x) \)[/tex] as the inverse of [tex]\( f(x) \)[/tex], applying [tex]\( h \)[/tex] after [tex]\( f \)[/tex] should revert [tex]\( f \)[/tex]'s transformation, returning the original [tex]\( x \)[/tex].
Thus, the value of [tex]\( h(f(x)) \)[/tex] is indeed:
[tex]\[
\boxed{x}
\][/tex]
