# CAT 2021 Set-2 | Quantitative Aptitude | Question: 15

375 views

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

1. $\left(\frac{3}{7}, \frac{1}{2} \right)$
2. $\left[\frac{3}{7}, \frac{1}{2} \right)$
3. $\left[\frac{3}{7}, \frac{8}{9} \right)$
4. $\left[\frac{4}{9}, \frac{8}{9} \right]$

Given that, $f(x) = \dfrac{x^{2} + 2x + 4}{2x^{2} + 4x + 9}; x \in \mathbb{R}$

Let $y = \dfrac{x^{2} + 2x + 4}{2x^{2} + 4x + 9}$

$\Rightarrow y (2x^{2} + 4x + 9) = x^{2} + 2x + 4$

$\Rightarrow 2yx^{2} + 4yx + 9y = x^{2} + 2x + 4$

$\Rightarrow x^{2} (2y-1) + x (4y-2) + 9y – 4 = 0$

For real values of $x,$ discriminant should be greater than equal to zero.

$\Rightarrow D \geq 0$

$\Rightarrow b^{2} – 4ac \geq 0 \quad [ \because \text{For} \; ax^{2} + bx + c = 0]$

$\Rightarrow (4y-2)^{2} – 4 (2y-1)(9y-4) \geq 0$

$\Rightarrow 16y^{2} + 4 – 16y – (8y – 4) (9y – 4) \geq 0$

$\Rightarrow 16y^{2} – 16y + 4 – (72y^{2} – 32y – 36y + 16) \geq 0$

$\Rightarrow 16y^{2} – 16y + 4 – 72y^{2} + 68y – 16 \geq 0$

$\Rightarrow – 56y^{2} + 52y – 12 \geq 0$

$\Rightarrow 56y^{2} – 52y + 12 \leq 0$

$\Rightarrow 14y^{2} – 13y + 3 \leq 0$

$\Rightarrow 14y^{2} – 7y – 6y + 3 \leq 0$

$\Rightarrow 7y (2y – 1) – 3 (2y – 1) \leq 0$

$\Rightarrow (2y – 1) (7y – 3) \leq 0$

$\boxed{ y \in \left[ \frac{3}{7}, \frac{1}{2} \right]}$

$\therefore$ The range of the function is $\left[ \dfrac{3}{7}, \dfrac{1}{2} \right).$

Correct Answer $: \text{B}$

Consider the pair of equations: $x^{2} – xy – x = 22$ and $y^{2} – xy + y = 34.$ If $x>y,$ then $x – y$ equals$7$$8$$6$$4 1 votes 1 answer 2 500 views If a rhombus has area 12 \; \text{sq cm} and side length 5 \; \text{cm}, then the length, \text{in cm}, of its longer diagonal is\sqrt{13} + \sqrt{12}$$\sqrt{37} +... 1 votes 1 answer 3 1,271 views The number of ways of distributing$15$identical balloons,$6$identical pencils and$3$identical erasers among$3$children, such that each child gets at least four ba... 1 votes 1 answer 4 437 views For a sequence of real numbers$x_{1}, x_{2}, \dots , x_{n},$if$x_{1} – x_{2} + x_{3} – \dots + (-1)^{n+1} x_{n} = n^{2} + 2n$for all natural numbers$n,$then the... 1 votes 1 answer 5 665 views For a real number$x$the condition$|3x – 20| + |3x – 40| = 20$necessarily holds if$9 < x < 14$$6 < x < 11$$7 < x < 1210 < x < 15\$