Question

Let the function $f:R\to R$ be defined by $f\left(x\right)=cosx,\forall x\in R$. Show that $f$ is neither one-one nor onto.

Answer 1

Given function, $f\left(x\right)=cosx,\forall x\in R$
Now, $f\left(\frac{\pi }{2}\right)=cos\frac{\pi }{2}=0$
$f(-\frac{\pi }{2})=cos(-\frac{\pi }{2})=0\Rightarrow f\left(\frac{\pi }{2}\right)=f(-\frac{\pi }{2})$

So, $f\left(x\right)$ is not one-one.

Now, from the definition of $f\left(x\right)$, the codomain of $f$ is $R$. However, the range of $f=[-1,1]$. Since the range of $f$ is a proper subset of its codomain, $f$ is not onto.