It is well known that for same type of functions, the composition also exhibits the same behavior (i.e., injective, surjective, bijective). Also, there is one property that links the functions' properties in the real analysis domain too: If a function $f$ is bijective, then it is monotonic iff $f^{-1}$ is monotonic.
I have a very simple example, as I am not aware of any way (in literature, or wiki or MSE) to find more about properties of compositions thereof.
If there are two functions such that one is bijective, and the other is injective, then what can be stated about the composition before-hand (i.e., without trying out the actual composition)
First example is $f(x) = x^2, g(x) = x+1$ over the domain of positive reals and co-domain of reals. Here, $g(x)$ is bijective, and $f(x)$ is injective. Also, the both compositions are injective only, as: $f(g(x)) = (x+1)^2, g(f(x)) = x^2 + 1$.
Second is of $f(x) = 2x, g(x) = x+1$ over the domain of integers. Here, $g(x)$ is bijective, and $f(x)$ is injective. Also, the two compositions are having same properties as the original one being the last in the composition , as: $f(g(x)) = 2(x+1), g(f(x)) = 2x + 1$. Here, $f(g(x))$ is only injective, while $g(f(x))$ is bijective.
So, what is the reason for the difference of compositions' properties in both examples.