To simplify the expression [tex]\( f(f^{-1}(x)) \)[/tex], understand the relationship between a function [tex]\( f \)[/tex] and its inverse [tex]\( f^{-1} \)[/tex]. By definition:
- [tex]\( f^{-1}(x) \)[/tex] is the inverse function of [tex]\( f(x) \)[/tex].
This means:
- If you apply [tex]\( f^{-1} \)[/tex] to some value [tex]\( y \)[/tex], and then apply [tex]\( f \)[/tex] to the result, you should get back [tex]\( y \)[/tex]. Mathematically, this can be represented as:
[tex]\[
f(f^{-1}(y)) = y
\][/tex]
In our specific question, we have the value [tex]\( x \)[/tex].
1. Apply the inverse function [tex]\( f^{-1} \)[/tex] to [tex]\( x \)[/tex]:
[tex]\[
z = f^{-1}(x)
\][/tex]
Here, [tex]\( z \)[/tex] is the value such that [tex]\( f(z) = x \)[/tex].
2. Next, apply the function [tex]\( f \)[/tex] to the result [tex]\( z \)[/tex]:
[tex]\[
f(z) = f(f^{-1}(x))
\][/tex]
Since [tex]\( z \)[/tex] is the value such that [tex]\( f(z) = x \)[/tex], we know:
[tex]\[
f(f^{-1}(x)) = x
\][/tex]
Hence, the simplified form of the expression [tex]\( f(f^{-1}(x)) \)[/tex] is:
[tex]\[
\boxed{x}
\][/tex]
