Let T: P2(R)--->P3(R) be a linear transformation defined by

T(f(x))= 2f'(x) + Integral from 0 to x of 3f(t)dt.

i know R(T)= span( T(1), T(x), T(x^2) ) = Span( (3x, 2 + 3/2x^2,4x+x^3)

But what i need to know is now to they go from T(1) to 3x

from T(x) to ....

Also if possible if you can tell me how to find the rank, nullity, and if is one-to one or onto will be a great help.

thanks alot.