Skip to main content
Physics Guru is an online education and competitive examinations preparation portal founded by Manish Verma, an IIT Madras alumnus, providing learning resources for competitive examinations.

The emphasis is on conceptual understanding, analytical thinking and problem solving rather than simply memorising formulas and standard methods.

The objective is to help students understand what they are learning and develop the ability to solve problems they have not seen before.

$f(x) = \frac{{{x^2} - 3x - 6}}{{{x^2} + 2x + 4}}$

Domain/Range/Onto/Into

Let $f:R \to R$ be defined by $f(x) = \frac{{{x^2} - 3x - 6}}{{{x^2} + 2x + 4}}$. Then which of the following statements is(are) correct?

(A) f is onto
(B) Range of f is $\left[ { - \frac{3}{2},2} \right]$
(C) f is into
(D) Domain of f is [-4, 0]

Solution

Let $y = \frac{{{x^2} - 3x - 6}}{{{x^2} + 2x + 4}}$

$\Rightarrow (y-1)x^2 +(2y+3)x +4y+6 = 0$

$\because x\in R$, $(2y+3)^2-4(y-1)(4y+6) \geq 0$ assuming above equation is quadratic equation that is when $y \neq 1$. However, we notice that when $y=1, x\in R$ so y = 1 will be accepted at the end.

$\Rightarrow -12y^2 +4y + 33 \geq 0 $

$ \Rightarrow  - \left( {y + \frac{3}{2}} \right)\left( {y - \frac{{11}}{6}} \right) \ge 0$

The wavy-curve for the expression $- \left( {y + \frac{3}{2}} \right)\left( {y - \frac{{11}}{6}} \right)$ is shown below.




Clearly, $y \in \left[ { - \frac{3}{2},\frac{{11}}{6}} \right]$. Note that y = 1 is already accounted for.

Since range $\left[ { - \frac{3}{2},\frac{{11}}{6}} \right]$ is not same as co-domain R, the function is into function. Hence, option (C).