Inverse of a multivalued function

Question

I try to get the inverse of a simple function like x^2,x≥0. Still, when I use InverseFunction Mathematica gives me back -(y^0.5) which is indeed an inverse, but for x≤0:

InverseFunction[#^2 &]

InverseFunction::ifun: Inverse functions are being used. Values may be lost for multivalued inverses.

-Sqrt[#1] &

I know I could just fix the problem putting a minus before InverseFunction. But is there a way to choose which of the inverse function I can get back, when I deal with a non-injective function like x^2? In other words, which is the option so that InverseFunction gives me back (y^0.5) instead fo -(y^0.5)? Thank you

Answer 1

You can use ConditionalExpression to control which branch InverseFunction uses:

InverseFunction @ Function[x, ConditionalExpression[x^2, x<0]]
InverseFunction @ Function[x, ConditionalExpression[x^2, x>0]]

Function[x, ConditionalExpression[-Sqrt[x], x >= 0]]

Function[x, ConditionalExpression[Sqrt[x], x >= 0]]