We all know that the strict monotonicity(non-decreasing/non-increasing) of a function is a sufficient enough condition for it to have an inverse function.
If \(f(x)\) is strictly monotonic,
has an inverse function such that,
Where, is the inverse of .
I started writing down the various functions whose inverse existed and proceeded to plot them on the same graph and invariably I found that the function and it's inverse were symmetric about .
Is this true for all functions whose inverse exists? Or just a few? If only few such functions exist, please tell me how to define such a class of functions.If all functions whose inverse exists show this property, then please post a proof for this, as I was unable find a valid proof with my current limited knowledge of math. You are free to use any mathematical analysis to prove it(just make it easy for me to understand).
I have placed this note into calculus assuming that calculus is required to prove this property. If you feel that it has to be placed under some other topic please let me know.