Question

Let $f:R\to R$ be defined by $f\left(x\right)=2+{\cos }^{2}x.$ Then $f$ is

[2 marks]

• A. one-one but not onto
• B. onto but not one-one
• C. both one-one and onto
• D. neither one-one nor onto

Answer 1

The correct option is D neither one-one nor onto

$f\left(x\right)=2+{\cos }^{2}x$
$f\left(0\right)=f\left(2\pi \right)=3$
Hence, $f$ is not one-one.

For all $x\in R,$ $0\le {\cos }^{2}x\le 1$
$\Rightarrow 2\le 2+{\cos }^{2}x\le 3$
Hence, range of $f$ is $[2,3]$
Here, range $\ne$ codomain
Hence, $f$ is not onto.