What does the phrase "invariant under f" mean?

Question

Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $f(x,y)=(\frac{x}{3}+11y^2,-2y)$.

Let $A=\{(3y^2,y)|y\in \mathbb{R}\}$.

Show that $A$ is invariant under $f$.

What is it asking?

Answer 1

See that

$f(3y^2,y)=(\frac{3y^2}{3}+11y^2,-2y)=(12y^2,-2y)=(3(-2y)^2,(-2y))$

which is obiviously in the set A.