Verify injectiveness of the function $f:\Bbb R\rightarrow \Bbb R$ satisfying the following: $f(x)=2x-(f(x))^2$

Question

Verify injectiveness of the function $f:\Bbb R\rightarrow \Bbb R$ satisfying the following:
$$f(x)=2x-(f(x))^2$$

I just could see that we have 2 possible values for $f(0)$: $0,-1$, nothing more...

Answer 1

If $f(x)=f(y)$, then $f(x)+f(x)^2=f(y)+f(y)^2$, and hence $2x=2y$ and hence $x=y$.

Answer 2

Hint:

Just write $f(x)=f(y)$ and use the functional relation.

Answer 3

$$f^2(x)+f(x)-2x=0$$ $$f(x)=\frac{-1\pm\sqrt{1+8x}}{2}$$