[Night Stalker]

Well, depending on typos you have two different situations:

based on you first post
f(x)=| e^2 for all x less than the natural log of 2

OR
| 2 for all x greater than or equal too ln(2).

f(x) is one of two constants.

based of the second post:
f(x)=|e^x ...... this is a variable function until x>= ln(2), then f(x) = 2

The other stuff you posted is just to show that the inverse function of ln(x) = e^x thats all don't confuse these concepts.

There are two types of inverses. Multiplicitave and Additive.

For multiplicative, x times its inverse = 1
For additive, x plus it's inverse = 0

For example 2 x ¹/₂ = 1
2 + (-2) = 0