I'm wondering how the domain affects the inverse function.
Taking the example:
The inverse is
Why is the inverse restricted to 0 while the other was restricted to -3? I see it on a graph and I understand that it must be because the function stops being a 1 on 1 at x=0 but I'm wondering how we figure that out with functions that are more complex such as a sinus function or something?
What if we do it the other way around and take g(f(x))? Then we get which must mean that x has to be equal to or larger than -3 as well? Or doesn't that matter?
As far as I remember, f(g(x))=g(f(x))=x for it to be a valid inverse.
Well...According to wolfram I'm wrong (sqrt(-4+3))^2-3 - Wolfram|Alpha
How is it that this is calculated? I would have thought that we started calculating inside the root to get -1, then attempt to take the root but we would end up with an imaginary number...But perhaps that imaginary number squared is equal to -1? (I'm not very familiar with complex numbers).