# About the image of an affine variety into a projective space.

Assume to have a morphism $$f: \mathbb{A}^n \to \mathbb{P}^m$$, I want to compute the ideal of the the projective closure of $$f(\mathbb{A}^n)$$. I think that in general the image of an affine variety trought a morphism $$f(x)=(f_1(x), \dots, f_m(x))$$ is $$f(X)= Z(y_0 -f_0(x), \dots , y_m-f_m(x))$$ where $$y_i$$ are the coordinates in the target.

In the above situation $$f(\mathbb{A}^n)= Z(x_0 -f_0(x), \dots , x_m-f_m(x))$$ and $$x \in \mathbb{A}^n$$ which is an affine algebraic set and for taking the projective closure I have to homogenize the $$x_i -f_i(x)$$. Then the ideal it will be the radical of the homogenized ideal obtained. Is my argument correct? Thank you for the answers!