While solving inverse function problems, I got confused in a part, like for any Inverse function to be defined, it must be one-one and onto, then in many questions why the codomain is given more than the Range as if we know that the codomain must needs to be equivalent to the Range for the Inverse function to be valid or defined, then why does they gave us the codomain different from the range?
Kindly help me solving this doubt.
It is a matter of definition only. In some books invertible functions have to be one to one and onto, so if $f\colon A\to B$, then $f^{-1}\colon B\to A$. But in some references the function only has to be one to one. and the inverse function is defined on the range of the original function.
It is important to note that the word "range" of a function f : X → Y used to always mean what is nowadays called the codomain. But since then its meaning has been distorted to sometimes mean codomain (i.e., Y) and sometimes mean the image im(f) of f.
Where im(f) = f(X) = {y ∊ Y | ∃ x ∊ X such that f(x) = y}.
Because of this unfortunate ambiguity, it is much better to always avoid using the word "range" unless its meaning is crystal-clear. (And even then.)
It's best to stick with the words "codomain" and "image".
