When Can an Inverse Function Exist?

You can't always have an inverse function (See previous post or previous but one or thereabouts).

A function is defined as any mapping which is one-one or many-one. This means that for any input value a unique output value is generated.

Something like y= x ² is a many-one function. 2² and -2² give the same value.

You can always draw a graph of a function.

A function CAN'T be a one-many mapping. It doesn't make a lot of sense to most of us to have some kind of operation which could generate loads of different answers for exactly the same input.

So.

If a function would generate a one-many correspondence, it follows that it's not a function. So a many-one function CANNOT have an inverse, because its inverse would be one-many (ie not a function at all).

Only one-one type functions have inverse functions.