Composition of two functions is constant

Question

Suppose $f$ and $g$ are such that $f(g(x)) = 1$. Does this imply that $g(f(x))$ is constant?

Answer 1

NO.

$$f(x) = \begin{cases} 1 & x \geq 0 \\ 0 & x < 0 \end{cases}$$

$$g(x) = x^2, \,\,\, \forall x \in \mathbb{R}$$

We then have $$f(g(x)) = f(x^2) = 1, \,\,\,\, \forall x \in \mathbb{R}$$ whereas $$g(f(x)) = f(x)^2 = \begin{cases} 1 & x \geq 0 \\ 0 & x < 0 \end{cases}$$

Answer 2

No, $\forall x \in I$ and $I' = f^{-1}(1) $ \ $ g(x)$ $$f(g(x)) = 1$$ $$f(g(f(x))) = 1$$ $$g(f(x)) \in f^{-1}(1) = g(x) \cup I'$$