There are a few mistakes. I am not sure why ou are writing r instead of x. YOU even wrote f(

x), meaning that f is in terms of x NOT r.

You completed the square correctly.

In finding the inverse, you did not move the -3/2 to the lhs correctly. You took the sqrt of the lhs correctly but only took the sqrt of some factors on the rhs.

Please try to fix those errors and report back.