Find bases and kernel of linear map

Find bases and kernel of linear map

Find bases for the image and the kernel of the linear map $f:P_2(x) \to
P_1(x)$ given by $f(p(x)) = p'(x).$ Based on your results, indicate
whether $f$ is injective or surjective.
This is my answer.
Ker f = p(x) subset P2 given p(x)=0 = p(x) subset P2 given d/dx p(x)=0 =
p0 subset R =R
Img f = q(x) subset P1 given q(x)=d/dx p(x), p(x) subset of P = P1