Question

Classify the following function $f\left(x\right)$ defined in $R\to R$ as injective, surjective, both or none

$f\left(x\right)=x^3+6x^2+11x+6$

• A. Surjective but not injective
• B. Injective but not surjective
• C. Neither injective nor surjective
• D. Both injective and surjective

Answer 1

The correct option is A Surjective but not injective

$f\left(x\right)=x^3+6x^2+11x+6=0$.
One one roots as $x=-1$

Hence
$f\left(x\right)=(x+1)(x^2+5x+6)$
$=(x+1)(x+3)(x+2)$

Hence
$f\left(x\right)=0$ has three roots $-1,-2,-3$

Hence y meets the x axis at three points.

Hence
The above function is surjective but it is not injective.