Does every algebra automorphism preserve augmentation ideal filtration?

Question

Let $A=\displaystyle\bigoplus_{n\geq0}A_n$ be a graded algebra and let the augmentation homomorphism $\varepsilon:A\to A_0$ be the projection. Define the augmentation ideal, denoted by $A_+$, to be the kernel of $\varepsilon$. Let $A_+^k$ to be the $k$th power of $A_+$ in the sense of ideal multiplication.

If $f:A\to A$ is a ring (not necessarily graded) automorphism, is it always true that $f(A_+^k)\subseteq A_+^k$?

Answer 1

$A=k[x],\epsilon(x)=0, f(x)=x+1,f^{-1}(x)=x-1$