Solve $g(x + \frac{1}{x}) = x^y + \frac{1}{x^y}$

Question

Let $y$ be a real number.

Find $g$ such that

$$g(x + \frac{1}{x}) = x^y + \frac{1}{x^y}$$

Is valid for all real $x$.

Answer 1

Setting $x=e^u$ one gets $$g(2\cosh(u))=2\cosh(u·y)$$ which resolves to $$ g(z)=2\cosh(y·\text{Arcosh}(z/2)) $$ for all $z\ge 2$