How is the notation of an inverse function represented mathematically?

The notation for an inverse function is represented as ( f^{-1}(x) ). This notation signifies that the function ( f^{-1} ) is the inverse of the original function ( f ). The inverse function essentially "reverses" the effect of the function ( f ). If ( f ) takes an input x and produces an output y, the inverse function ( f^{-1} ) takes that output y and returns the original input x.

This relationship can be mathematically expressed as follows: if ( f(x) = y ), then ( f^{-1}(y) = x ). This concept is crucial in functions and their inverses, as it allows us to find original values from results.

The other forms presented do not correctly represent an inverse function. For example, ( f'(x) ) denotes the derivative of the function, ( f^2(x) ) indicates the composition of the function with itself, and ( f(x)-1 ) simply means subtracting 1 from the output of the function. None of these notations indicate the concept of reversing a function’s operation.