Question

$f:R\to R$ defined by $f\left(x\right)=\cos (2x+3)$ is

• A. injective only
• B. surjective only
• C. bijective
• D. neither injective nor surjective

Answer 1

Answer: (D)

neither injective nor surjective

Given, $f:R\to R$ defined by

$f\left(x\right)=\cos (2x+3)$

For injective(one-one):

$f\left(x_1\right)=f\left(x_2\right)\Rightarrow \cos (2x_1+3)=\cos (2x_2+3)$
$\Rightarrow 2x_1=2\pi +2x_2$
$\Rightarrow x_1=\pi +x_2$
$\therefore f\left(x_1\right)=f\left(x_2\right)\ne >x_1=x_2$
$\therefore$ $f$ is not injective.

For surjective(onto):
As $\cos (2x+3)$ lies between $-1$ and $1$
$\therefore$ co-domain $\ne$ range
$\therefore f$ is not surjective.