<s>[INST] $y =f(x) =(ax^2 + bx +c)/(dx^2+ex+f)$ We have to find the conditions for this it takes all real values. $$
y=f(x)=\frac{ax^2+bx+c}{dx^2+ex+f}
$$
We have to find the conditions for this it takes all real values.
MY solution
One approach is to equate it to y and for a quadratic of x and put discriminant greater than equal to 0..That is very lengthy.Is their any better method..?
 [/INST] $$f\left( x \right) =\frac { a{ x }^{ 2 }+bx+c }{ d{ x }^{ 2 }+ex+f } =\frac { a{ x }^{ 2 }+bx+c }{ d\left( { x }^{ 2 }+\frac { e }{ d } x+\frac { f }{ d }  \right)  } =\frac { a{ x }^{ 2 }+bx+c }{ d\left( { x }^{ 2 }+\frac { e }{ d } x+\frac { e^{ 2 } }{ 4{ d }^{ 2 } } -\frac { e^{ 2 } }{ 4{ d }^{ 2 } } +\frac { f }{ d }  \right)  } =\\ =\frac { a{ x }^{ 2 }+bx+c }{ d{ \left( x+\frac { e }{ 2d }  \right)  }^{ 2 }+\frac { 4fd-{ e }^{ 2 } }{ 4{ d } }  } $$
so range is $$R\setminus \left\{ -\frac { e }{ 2d }  \right\} \\ $$
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