Define powerset P(f) (P(real numbers) I'd like some help for clarification, as I have no professional help to ask (and also wouldn't want to pay for it yet).
This is part of a German book on mathematical analysis, I don't want the solution to this, as I want to figure it out for myself as much as possible - so please consider this, no matter how little I provide.
I have been given the same definitions as this (since it is the same sub-chapter):
power set of an identity idx is equal to the identity id of powerset x
The author now presents a final task to finish the chapter, and I'd love to actually do it without looking at the solution in the back.

Task:$$ \\ $$ Given that $$ f:\mathbb{R}  \rightarrow \mathbb{R} \\ \text{with} \ x \rightarrow f(x) \\ \ f(x) = x^2. \\ \ \\ \ \\ \text{Determine} \ P(f)(P(
 \mathbb{R} )) $$

Now I know that per definition of $$ P(f)(A)=f(A) $$
I can say that
$$
P(f)(P(\mathbb{R}))=f(P(\mathbb{R}))
$$
Now, I am quite confused on what the actual task is and what "determine" means. Is the solution just a mostly verbal "tell me what the powerset of this is" or is it rather asking me to define in a formal way..? 
 A: I'm glad to see you gave MathJax a try :D
"Determine [a set]" means to characterize the elements of the set so you can "easily" find out if a given explicit element is in the set or not. It is kind of a vague exercise formulation, basically a "Prove that $A=B$" without giving you $B$, but letting you find a suitable $B$ yourself. So basically, there is no "the right answer", only "the answer the author had in mind" and it is basically your goal to get as close as possible.
Example: When you are given $(x_1,y_1)-(x_2,y_2):=(x_1-x_2,y_1-y_2)$, $||(x,y)||:=x^2+y^2$ and $$U_r((x,y)):=\{(x_1,y_1):||(x_1,y_1)-(x,y)||\leq r^2\}$$
for $x,x_1,x_2,y,y_1,y_2,r\in\mathbb{R}$, and I say "Determine $U_r((x,y))$.", then a valid answer would be
$$U_r((x,y))=\{(x_1,y_1):(x_1-x)^2+(y_1-y)^2\leq r^2\}$$
Bonus points if you notice that $U_0((x,y))=\{(x,y)\}$ and $U_r((x,y))=\{\}$ if $r<0$. (For $r>0$ $U_r((x,y))$ describes the disk with center $(x,y)$ and radius $r$ in $\mathbb{R}^2$.
So in your case, you are basically expected to use the definitions again and again until there is nothing more to do. Any hint or solution would be in your book as well, so I'm leaving it out here.
EDIT: Oh wait, we have spoilers, so here is what I would do :D
First let me note that I believe your first step is wrong.
$$P(f)(P(\mathbb{R}))=\{P(f)(A):A\in P(\mathbb R)\}$$
This is important: Since $P(\mathbb{R})\notin P(\mathbb{R})$ but $P(\mathbb{R})\subseteq P(\mathbb{R})$, you don't get to use the definition of $P(f)$ but the definition of $f(A)$. Of course it could be that the author made some remark about notation, so that your first step is actually right an I am wrong. You can only find that out by looking at the solution later.
Rest is spoiling.

 Since $P(f)(A)= f(A) = \{x^2:x\in A\}$ we have
 $$P(f)(P(\mathbb{R}))=\{\{x^2:x\in A\}:A\in P(\mathbb{R})\}$$
 Now any nonnegative number $y$ has a nonnegative number $x$ with $y=x^2$ ($x=\sqrt{y}$), so for each $B\in P(\mathbb{R}_{\geq 0})$ there is a $A\in P(\mathbb{R})$ with $B=\{x^2:x\in A\}$. On the other hand, since $x^2$ is never negative, there are no other kinds of sets in $\{\{x^2:x\in A\}:A\in P(\mathbb{R})\}$. We conclude
 $$P(f)(P(\mathbb{R}))=\{B:B\in P(\mathbb{R}_{\geq 0})\}=P(\mathbb{R}_{\geq 0})$$

