Is there a family of functions $g(x)$ or a form for g in which the following statement
$$g(x^y + z) = g(x)^y +z$$
is verified, with {x,y,z} in $\mathbb{R}$?
Disclaimer: This question came to my mind without any background so I'm not sure if it has a proper solution or not.
As @Kavi Rama Murthy said you must suppose $x \geqslant 0$. Then by setting $x=1$ and $c=g(1)-1$, you have $g(z)=c+z$. Now continue by what @TokenToucan said and you have $c=c^2+2cx$, that must be satisfied for any $x$, which results in $c=0$, equivalents to $g(1)=1$, and in general $g(x)=x$. Notice that at the end there is no restriction like $x \geqslant 0$, and $x$ can get any real value.