Question

what is the inverse of y=2x^2-3x+1

Answer 1

the function is not 1 to 1 so it doesn't have an inverse

Answer 2

I think you are asking for the inverse "relation" not a function.

if you do the horizontal line test, we can see that there are many points that intersect more than twice, so it is not an "inverse function"

Answer 3

but if you are asking for an "inverse relation"
all you have to do is to switch the x's with the y's and
start solving for y.

you should get a positive and a negative version of a root function.
This fact indicates that it is not a "function."
So as a "relation" it is ok.