Natural invertible functions Can you give examples of two functions f: N -> N and g: N -> N that are invertible in which g is not injective? And how did you proceed to find them? Also, is f(x) = x^2 [0, ∞[ a real or natural function?

Is the answer will be the same if it only said (g o f) = iN and g is not injective? forget about the invertible
Consider the function $f(x)=2x$ mapping $\mathbb{N}\mapsto \mathbb{N}$.  Clearly, $f(x)$ is one-to-one and hence invertible.  Note that the range of $f$ is just the even numbers.  $f$ is injective (not surjective).

Take $g(x)=x/2$ if $x$ is even, and $g(x)=1$ if $x$ is odd.  Since $g$ maps many values to 1, $g$ is not injective, and hence, not invertible.  $g$ is surjective.  $g$ is not injective.

But, $g(f(x))=x$ for any $x\in \mathbb{N}$.  So, $g(f(x))$ is invertible.  In fact, $g(f(x))$ is both injective and surjective.
Consider the function $f(x)=2x$ mapping $\mathbb{N}\mapsto \mathbb{N}$.  Clearly, $f(x)$ is one-to-one and hence invertible.  Note that the range of $f$ is just the even numbers.  $f$ is injective (not surjective).

Take $g(x)=x/2$ if $x$ is even, and $g(x)=1$ if $x$ is odd.  Since $g$ maps many values to 1, $g$ is not injective, and hence, not invertible.  $g$ is surjective.  $g$ is not injective.

But, $g(f(x))=x$ for any $x\in \mathbb{N}$.  So, $g(f(x))$ is invertible.  In fact, $g(f(x))$ is both injective and surjective.