> 
> Let $f: [0, \infty) \rightarrow [0,2)$ be defined by $$f(x) = \left\{
> \begin{array}{ll}
>  \frac{1}{x} + 1, & x > 1 \\
>  x, & x\le 1 \\
> \end{array} 
> \right.$$
> 
> 
> The question asks to prove that the function is bijective.
> 
> 
> 

I have to prove it is injective and surjective. 

Injectivity:  

if $x,y<1$, then $x=y$  

if $x,y>1$ then, $\frac{1}{x}+1=\frac{1}{y}+1$, then $x=y$

> 
> I get this bit, but I don't get how to prove the surjective part of the proof. 
> 
> 
> 

Please could you help?