Prove that $f(f(x)) - f(x) = x +1$, is one to one Given:
 $Df = \mathbb{R}$

And, $f(f(x)) - f(x) = x + 1$

Prove that $f$ is inverse. I've busted my head with this one. Any help is appreciated
Let $f(x)=f(y)$. Then $x+1 = f(f(x)) - f(x) = f(f(y))-f(y) = y+1$ and so $x=y$.
Restriction: Here we assume that we are looking for functions nice enough to have power series expansions of some radius of convergence around every point.

Then we rewrite $$f(x) = 1+x+f(f(x))$$

Now it is trivial to see that $f(f(x))$ will expand into a higher order polynomial for any truncation to an order >1 we can make. So $f(x)$ must be the equation of a line: 

$$f(x) = a+bx$$

A linear function is inversible.

( With risk of sounding arrogant, from an engineering perspective all interesting smooth-enough functions are like this anyways. )