Part 2/6:
The concept of an inverse function arises when we have a function, denoted as ( y = f(x) ). The inverse of this function is represented as ( f^{-1}(x) ). What does this notation mean? It signifies that if we have a function where ( y ) is dependent on ( x ), then in the inverse function, ( x ) becomes dependent on ( y ). In essence, we are swapping ( x ) and ( y ).
It’s important to clarify that ( f^{-1}(x) ) does not imply that we are raising ( f(x) ) to the power of negative one or taking a derivative—these are distinct concepts.
Example: Finding an Inverse Function
To illustrate the concept further, let’s consider a function given by the equation ( y = x^3 + 2 ). To find the inverse, we can follow these steps: