<s>[INST] Finding the inverse function of: $y=3x^2-6x+1$ I am trying to find the inverse function, $f^{-1}(x)$ of this function, $y=3x^2-6x+1$ 
Normally the teacher told us to interchange the $x$ and $y$
$x=3y^2-6y+1$ and... now transpose the formula to make $y$ the subject. But this look not possible. Is there another alternative method to find the $f^{-1}(x)?$
 [/INST] $y=3x^2-6x+1=3(x^2-2x)+1$;
$y=3(x-1)^2 -3+1=$
$3(x-1)^2-2$;
A parabola opening upwards, vertex $(1,-2)$;
Local inverse for:
1) $x\gt 1$: 
$(1/3)(y+2)=(x-1)^2$;
$x= 1+\sqrt{(1/3)(y+2)}$;
Switch:
$y=1+\sqrt{(1/3)(x+2)}$, $x \gt -2$.
Local inverse for
2) $x \lt 1$;
$-\sqrt{(1/3)(y+2)}=x-1$;
Switch:
$y=1-\sqrt{(1/3)(x+2)}$, $x \gt -2$.
Does a global inverse exist? 
Is there an inverse function for $(-\epsilon+1, \epsilon +1)$, $\epsilon >0$ ?
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