<s>[INST] Proving that this function is not injective Let $f:\mathbb{R}^2\to\mathbb{R}^2$ given by $f(x,y)=((x+2)^2+y, 2x-3y-1)$ 
how do I prove that this function is not injective and thus not globally invertible?
I tried with $f(x,y)=f(u,v)$ but the system is quite impossible/hard for me.. is there another way?
 [/INST] Find a case where $((x+2)^2 +y,2x -3y -1) = ((w+2)^2+ u, 2w-3u-1)$ but $(x,y)\ne (w,u)$.
Just spitballing:
If $x =0$ then we need
$(0+2)^2 + y = (w+2)^2 + u$ or 
$y+4 = w^2 + 4w + 4 + u$ or $y = w^2 + 4w+u$
And $-3y-1 = 2w-3u-1$ so
$y = u-\frac 23w = w^2 + 4w + u$ so
$w^2 + \frac {14}3w=0$  If $w\ne 0$ then
$w=-\frac {14}3$.
$y = \frac {28}9 + u$ and $u$ can be anything.
So $f(0,\frac {28}9) = (2^2+\frac {28}9, -\frac {28}3-1) = (\frac {64}9,-\frac {31}3)$
And $f(-\frac {14}3,0) = ((-\frac {8}3)^2 + 0, -\frac {28}2-1)=(\frac {64}9, -\frac {31}3)$.
Thus it is not injective.
Probably more efficient ways to solve.
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