Finding the domain interval of a reverse function

Question

I have a question about finding the domain interval of a reverse function. If you have a function $f$ and its domain is $(-∞;-1]∪[1;+∞)$, how do you find its reverse function's domain?

Answer 1

That depends upon the definition of inverse function (I suppose that that's what you mean when you talk about “reverse”) that you use:

• if we say that the inverse of a function $f\colon A\longrightarrow B$ is a function $g\colon B\longrightarrow A$ such that $g\circ f=\operatorname{Id}_A$ and $f\circ g=\operatorname{Id}_B$, then the answer is the codomain of your function;
• if we say that the inverse of a function $f\colon A\longrightarrow B$ is a function $g\colon f(A)\longrightarrow A$ such that $g\circ f=\operatorname{Id}_A$ and $f\circ g=\operatorname{Id}_{f(A)}$, then the answer is the range of your function, that is, $f\bigl((-\infty,-1]\cup[1,+\infty)\bigr)$.