How do I approach this problem? My book gives answer as $[\frac{3-\sqrt{5}}{2},\frac{3+\sqrt{5}}{2}]$. I tried forming an equation in $y$ and putting discriminant greater than or equal to zero but it didn't work. Would someone please help me?
I get $x^2 (y-1) + 2x (y+1) + (5y-5) =0$ and discriminant gives $2y^2 - y + 2 \leq 0$, which has complex roots.
Hint: You will get $$(y+1)^2\geq 5(y-1)^2$$ for $$y\neq 1$$ Can you solve this?
Hint:
The derivative of $\dfrac{x^2-2x+5}{x^2+2x+5}$ is $\dfrac{4 (x^2 - 5)}{x^2+2x+5}$ and so the critical points are $\pm \sqrt 5$.
Consider also $\displaystyle\lim_{x\to\pm\infty}\dfrac{x^2-2x+5}{x^2+2x+5}=1$.
Since $$y = \frac{x^2 - 2x + 5}{x^2 + 2x + 5}$$ we obtain \begin{align*} y(x^2 + 2x + 5) & = x^2 - 2x + 5\\ yx^2 + 2yx + 5y & = x^2 - 2x + 5\\ (y - 1)x^2 + (2y + 2)x + 5(y - 1) & = 0\\ (y - 1)x^2 + 2(y + 1)x + 5(y - 1) & = 0 \end{align*} which is equivalent to your equation $x^2(y - 1) + 2x(y + 1) + (5y - 5) = 0$. However, you should have obtained the discriminant \begin{align*} \Delta & = b^2 - 4ac\\ & = [2(y + 1)]^2 - 4(y - 1) \cdot 5(y - 1)\\ & = 4(y^2 + 2y + 1) - 20(y - 1)^2\\ & = 4y^2 + 8y + 4 - 20(y^2 - 2y + 1)\\ & = 4y^2 + 8y + 4 - 20y^2 + 40y - 20\\ & = -16y^2 + 48y - 16 \end{align*} We want $\delta \geq 0$, so \begin{align*} 0 & \leq -16y^2 + 48y - 16\\ 16y^2 - 48y + 16 & \leq 0\\ y^2 - 3y + 1 & \leq 0\\ y^2 - 3y & \leq -1\\ y^2 - 3y + \frac{9}{4} & \leq \frac{5}{4}\\ \left(y - \frac{3}{2}\right)^2 & \leq \frac{5}{4}\\ \left|y - \frac{3}{2}\right| & \leq \frac{\sqrt{5}}{2} \end{align*} Thus, $$-\frac{\sqrt{5}}{2} \leq y - \frac{3}{2} \leq \frac{\sqrt{5}}{2} \implies \frac{3 - \sqrt{5}}{2} \leq y \leq \frac{3 + \sqrt{5}}{2}$$ so the range of the function is $$\left[\frac{3 - \sqrt{5}}{2}, \frac{3 + \sqrt{5}}{2}\right]$$