what is your question?
First of all that distribution function is wrong.
It's ok that F(x)=0 when x is less than 1
and F(x)=1 when x is greater than 2
BUT it cannot be x from 1 to 2.
MAYBE your instructor meant F(x)=x-1, since F cannot exceed 1.
This says that F(1.5)=1.5 which means that which is a no-no.
BUT in order to transform, I use calc 1
which can easily be proved by differentiating the cdfs, which was the approach of your teacher.