Domain and Range

Given, $$f\left(x\right)=\dfrac{3x-11}{x^2-6x+14}=y$$ (say)

So, $$\dfrac{3x-11}{x^2-6x+14}=y$$

or, $$3x-11=y\left(x^2-6x+14\right)$$

or, $$3x-11=yx^2-6xy+14y$$

or, $$yx^2-6xy-3x+14y+11=0$$

or, $$yx^2-\left(6y+3\right)x+\left(14y+11\right)=0$$

Now, $$x$$ has to be real, so discriminant of this equation has to be greater than or equal to zero.

So, $$D\ge0$$

or, $$\left(6y+3\right)^2-4y\left(14y+11\right)\ge0$$

or, $$36y^2+36y+9-56y^2-44y\ge0$$

or, $$-20y^2-8y+9\ge0$$

or, $$20y^2+8y-9\le0$$

or, $$20y^2+18y-10y-9\le0$$

or, $$2y\left(10y+9\right)-1\left(10y+9\right)\le0$$

or, $$\left(2y-1\right)\left(10y+9\right)\le0$$

or, $$-\dfrac{9}{10}\le y\le\dfrac{1}{2}$$