<s>[INST] Range of a Rational Function How to find the Range of function  $$f(x)= \frac{x^2-3x-4}{x^2 - 3x +4}$$
I tried to equate the expression to $y$, then cross multiplied 
$$ y= \frac{x^2-3x-4}{x^2 - 3x +4}$$
$$ y(x^2 - 3x +4)= x^2-3x-4 $$
bought the terms to one side so it becomes a quadratic and made Discriminant to zero , but i cant seem to reach anywhere.. Any suggestions? 
 [/INST] Given $$\displaystyle y = \frac{x^2-3x-4}{x^2-3x+4}\Rightarrow yx^2-3yx+4y = x^2-3x-4$$
so we get $$\displaystyle (y-1)x^2-3(y-1)x+4(y+1) = 0$$
For real values of $y\;,$ Then given equation has real  roots. So $\bf{Discriminant\geq 0}$
So $$9(y-1)^2-16(y-1)\cdot (y+1)\geq 0$$
So we get $$(y-1)\left[9(y-1)-16(y+1)\right]\geq 0$$
so $$(y-1)\left[-7y-25\right]\geq 0$$
So $$\displaystyle (y-1)(y+\frac{25}{7})\leq 0$$
So $$\displaystyle -\frac{25}{7}\leq y\leq 1\Rightarrow y\in \left[-\frac{25}{7}\;,1\right]$$
Now when $y=1\;,$ Then $$\displaystyle 1 = \frac{x^2-3x-4}{x^2-3x+4}\Rightarrow -4=+4\; \bf(Not\; Possible)$$
So  we get $y\neq 1$
So Original Range $$\displaystyle -\frac{25}{7}\leq y< 1\Rightarrow y\in \left[-\frac{25}{7}\;,1\right)$$
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