A similar question already exists asking how to divide two quadratic functions, but it is solved by factoring and using the greatest common denominator.
If I have $f(x)=\frac{3x^2-2x+2}{x^2+2x+2}$, What is the domain and range of this function?
What I am trying to do is find the domain of the inverse function to solve for the range, but I cannot find the inverse function. So far, I have $x=\frac{3y^2-2y+2}{y^2+2y+2}$ and I need to solve for y in terms of x. How can I do so?
Writing $y$ for $f(x)$, render
$3x^2-2x+2=y(x^2+2x+2)$
Then make a quadratic equation for $x$:
$(y-3)(x^2)+(2y+2)(x)+(2y-2)=0$
The only values of $y$ that can be reached with real values of $x$ are those where the discriminant is nonnegative:
$(2y+2)^2-4(y-3)(2y-2)\ge0; -4y^2+40y-20=-4(y^2-10y+5)\ge0$
The limits of the range are then the roots of $y^2-10y+5=0$. Since the quadratic coefficient is negative, the values of $y$ that give positive discriminant and thus real values for $x$ lie between rather than beyond these limits.