CAT 2021 Slot 2 QA Question 4: For all real values of $x$, the range of the function $f(x)=\dfrac{x^2+2x+4}{2x^2+4x+9}$ is

Question

For all real values of $x$, the range of the function $f(x)=\dfrac{x^2+2x+4}{2x^2+4x+9}$ is

Accepted Answer

Let $y = f(x)$. Then
$y=\dfrac{x^2+2x+4}{2x^2+4x+9}$
$y(2x^2+4x+9)=x^2+2x+4$
$(2y-1)x^2+(4y-2)x+(9y-4)=0$

For some real $x$ to exist, this quadratic in $x$ must have non-negative discriminant.

$(4y-2)^2-4(2y-1)(9y-4) \geq 0$
$-56y^2+52y-12 \geq 0$
$14y^2-13y+3 \leq 0$
$(7y-3)(2y-1) \leq 0$
$\dfrac

For all real values of xxx, the range of the function f(x)=x2+2x+42x2+4x+9f(x)=\dfrac{x^2+2x+4}{2x^2+4x+9}f(x)=2x2+4x+9x2+2x+4​ is