To solve for [tex]\( x \)[/tex] in the equation [tex]\( f\left(f^{-1}(x)\right) = x \)[/tex], let's follow these steps in detail:
1. Understanding the Function and Its Inverse:
- The symbol [tex]\( f^{-1} \)[/tex] denotes the inverse function of [tex]\( f \)[/tex].
- By definition, applying a function [tex]\( f \)[/tex] to its inverse [tex]\( f^{-1} \)[/tex] should yield the original input [tex]\( x \)[/tex].
- Mathematically, [tex]\( f(f^{-1}(x)) = x \)[/tex].
2. Interpreting the Equation:
- The equation [tex]\( f(f^{-1}(x)) = x \)[/tex] means that if you take the inverse function of [tex]\( x \)[/tex] and then apply the function [tex]\( f \)[/tex] to that result, you will end up with [tex]\( x \)[/tex].
3. Solving for [tex]\( x \)[/tex]:
- When you apply the function [tex]\( f \)[/tex] to [tex]\( f^{-1}(x) \)[/tex], the output is simply [tex]\( x \)[/tex].
- This is a fundamental property of functions and their inverses and holds true for any [tex]\( x \)[/tex].
Therefore, the solution to the equation [tex]\( f\left(f^{-1}(x)\right) \)[/tex] is:
[tex]\[ x \][/tex]
