# What is the $2-norm$ relative Condition Number of $f$ with respect to pertubations of x. If $f(x)=||x||_2^2$ for $x \in \mathbb{R}$,

if $$f(x)=||x||_2^2$$ for $$x \in \mathbb{R}$$, What is the $$2-norm$$ relative condition number of $$f$$ with respect to pertubations of x?

What I have tried:

I have tried the following formula from trefethen book page 131-Numerical Linear Algebra.

I used the 2-norm for: ||J||, ||f(x)||, $$||x||$$. And my final answer was $$\kappa = 2$$

My question is. Is this process correct? and how do I know wich norm I have to use for each one of the terms $$||J(x)||, ||f(x)|| and ||x||$$.

I have these questions because in the same book page 89 and equation (12.3) they state:

So, I want to know if using the same 2-norm for all the expression is correct and when it is not correct. For instance, in the following example from the same book, page 91 different norms are used.