Question 22
(UKR) (a) Do there exist functions $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ such that
$$f(g(x))=x^2\quad\text{and}\quad g(f(x))=x^3\quad\text{for all }x\in\mathbb R?$$
(b) Do there exist functions $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ such that
$$f(g(x))=x^2\quad\text{and}\quad g(f(x))=x^4\quad\text{for all }x\in\mathbb R?$$
$$f(g(x))=x^2\quad\text{and}\quad g(f(x))=x^3\quad\text{for all }x\in\mathbb R?$$
(b) Do there exist functions $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ such that
$$f(g(x))=x^2\quad\text{and}\quad g(f(x))=x^4\quad\text{for all }x\in\mathbb R?$$