doubt

u shud draw the graph.
First find the domain ull get an interval (1/2,2)

Now the function y=5x-2-2x² will be symmetric about the point x= -5/2(-2) = 5/4

So the interval in which the function has inverse is:
(1/2,5/4] OR [5/4,2)

k....i didnt understand third step..........

and y dont we check surjectivity over here find existance of inverse??

as it is not given what the interval of the range is taken, u dont need to take surjectivity into account. Basically the answer given above is the complete set. there can be subsets of that also where the function has an inverse taking appropriate domains and range