Getting inverse of x^2 with positive x

Question

My goal it to get inverse of x^2 for positive x. But the result shows always minus -Sqrt[y]. How to correct

u[x_] := x ^2
$Assumptions = x > 0;
Refine[InverseFunction[u][y], y \[Element] Reals]

I have tried this code also but is also shows -Sqrt[y].

Refine[x /. Solve[y == u[x], x][[1]], y \[Element] Reals]

Answer 1

Using this example from the documentation as a guide:

if = InverseFunction[Function[x, ConditionalExpression[u[x], x>0]]]

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

Check:

if[y]

ConditionalExpression[Sqrt[y], y >= 0]

With actual numbers you get:

if[2.]
if[-2.]

1.41421

Undefined