Is $f(x, y) := {1 \over x^2 + y^2}$ function with a norm?

Question

Let

$f: \Bbb R^2 \rightarrow \Bbb R$,

$$f(x, y) := {1 \over x^2 + y^2}$$ for $(x, y) \in \Bbb R^2 \setminus \{0\}$, $0$ otherwise.

In of my books, it is claimed that this would be a function of the type

$$\Bbb R^n \rightarrow \Bbb R,$$

$$x \rightarrow f(||x||),$$

but why is that the case? $||x||$ denotes a norm, right? But what kind of norm does $x^2 + y^2$ denote?

Answer 1

Hint. One may take the euclidian norm $$ ||x||=\sqrt{x^2 + y^2}=(x^2 + y^2)^{1/2}. $$ Then can you identify $f$?