Polytime Mapping Reduction from Language A to Language A (identity)

Question

How would I create a polytime mapping reduction to prove A ≤p A for any language A.

I was thinking to assume A is in P to start. For every 𝑥: 𝑥∈𝐴 iff 𝑓(𝑥)∈𝐴.

But I am not sure what to do from there, any advice? I don't get how to show that you can polytime reduce a function to itself, because if you assume A has a Turing machine and a polytime decider, a function that is a polytime mapper, then how do you show its equal to each other?

Answer 1

You need to construct a polytime function $f$ with the following property:

For every $x$: $x \in A$ iff $f(x) \in A$.

You take it from here.