Given function $f:\Bbb R\to\Bbb R:f(x)=\cos x$, check which properties it has

Question

Given function $f:\mathbb{R}\to \mathbb{R}:f(x)=\cos x$, check whether it is

1. surjective
2. injective
3. increasing
4. decreasing
5. strictly increasing
6. strictly decreasing

My Idea:

$f(0)=f(2\pi)$ but $0 \neq 2\pi$ this f is not one one

consider $y=2 \in \mathbb{R}$

There does not exist any $x\in \mathbb{R}$ such that $f(x)=\cos x=y=2$

then $f$ is not onto

what about other options

Answer 1

Note that $f'(\pi/2)=-1<0$ and $f'(-\pi/2)=1>0$. $f$ is thus not increasing, decreasing, strictly increasing or strictly decreasing.

Indeed, it has none of the six listed properties.

Answer 2

Infact if it is strictly increasing or decreasing then function must be one one because we can do ordering between real numbers .