CAT 2025 Slot 3 QA Question 11: For real values of $x$, the range of the function $f(x) = \dfrac{2x - 3}{2x^{2} + 4x - 6}$ is

Question

For real values of $x$, the range of the function $f(x) = \dfrac{2x - 3}{2x^{2} + 4x - 6}$ is

Accepted Answer

Let $y = f(x)$. Then
$y = \dfrac{2x - 3}{2x^{2} + 4x - 6}$
$y(2x^{2} + 4x - 6) = 2x - 3$
$2yx^{2} + (4y - 2)x + (3 - 6y) = 0$
For real $x$, this quadratic in $x$ must have discriminant at least $0$.
$(4y - 2)^{2} - 4(2y)(3 - 6y) \geq 0$
$64y^{2} - 40y + 4 \geq 0$
$16y^{2} - 10y + 1 \geq 0$
$(8y - 1)

For real values of xxx, the range of the function f(x)=2x−32x2+4x−6f(x) = \dfrac{2x - 3}{2x^{2} + 4x - 6}f(x)=2x2+4x−62x−3​ is

For real values of $x$ , the range of the function $f(x) = \dfrac{2x - 3}{2x^{2} + 4x - 6}$ is