Help with parabolic function inverses

1)Why do you use a certain half of a parabola when graphing the inverse?
2)How do I know which side to use when I want to take the inverse of a parabolic function?
3)Why does the parabola have to be split in the first place, because if the original function is just reflected by the line y=x shouldnt the inverse function map the range values to the domain?

1&3) seem to be asking the same thing. There is no need to split the parabola in order to find its inverse, unless you want its inverse to be a FUNCTION. A function has no repeated x-values, so its inverse has no repeated y-values, which a parabola has before we split it.

2) It doesn't matter which side you use. Once it is one-to-one (all unique x & y values) then it will have an inverse function.

1st : an inverse only exists for functions which are one to one...and a parabola is not.
2nd : there can be an infinite number of inverses coming from a one to many function ;
for a parabola typically you specify the domain being used so that the function is one to one ;

ex : y = x² ,x in [ - 2 , 2 ] ; for inverse domain [0,1) use function domain [ 0 , 1 ) ;
for inverse domain [ 1 , 2 ) use function domain (-√ 2, -1 ] , inverse domain [2 , 3 )
use function domain [√2 , √3) , for inverse domain [3 , 4 ] use function domain (√3 , 2 ] ;
geometrically this is sketch y = x² , y = 1 y = 2 , y = 4 , y = 4... to generate a one to one function
we 1st use the right side , then the left , back to the right , finishing on the left.This new function has an inverse which you can graph by reflection across y = x !

1. symmetric to origin
2. mostly, problems are given with ranges such as [1,4] or other
since we know that it's symmetric, it's just a matter of changing +/-.
ex)
y=x^2+91
to find inverse
y-91=x^2
+-sqrt(y-91)=x
y=x+-sqrt(x-91)
3.
to make thinking process easier