Example where $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective Can anyone come up with an explicit example of two functions $f$ and $g$ such that: $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective?

I tried the following: $$f:\mathbb{R}\rightarrow \mathbb{R^{+}}
$$ $$f(x)=x^{2}$$

and  $$g:\mathbb{R^{+}}\rightarrow \mathbb{R}$$
$$g(x)=\sqrt{x}$$

$f$ is not injective, and $g$ is not surjective, but $f\circ g$ is bijective

Any other examples?
The simplest example: 


$X=\{0\},Y=\{0,1\}$, $f(0)=0$, $g(0)=g(1)=0$. 


(Here $g\circ f$ is the bijection, since I inadvertently reversed the names of the functions.)

Everything else is an elaboration of one of this idea.
X={0},Y={0,1}, f(0)=0, g(0)=g(1)=0.
more info could be found at here.