Edges of rectangle mapped to a new plane

Question

If a rectangular plane lies in quadrant I and is transformed by a polynomial function, will the edges of the rectangle always map to the edges of the transformed plane?

Will all points along the edges of the rectangle map to outer points on the new plane?

If not, what is a counter example?

For clarification, I'm asking if its possible if a point INSIDE the body of a plane could map to an edge point in the new plane.

Answer 1

Suppose the original rectangle is $[-1,1] \times [-1,1]$ and the polynomial transformation is $(x,y) \mapsto (x^2,y^2)$.