$Iff:R\to Cisdefinedbyf\left(x\right)=e^{2ix}\forall x\in R,thenfis\left(whereCdenotesthesetofallcomplexnumbers\right)$

$\begin{array}{l}f\left(x\right)=e^{2ix}=\cos 2x+i\sin 2x\\ weknowthat\cos xand\sin xareperiodicfunction\\ \sin 2xand\cos 2xperiod=\frac{2\pi }{\pi }=\pi \\ \therefore f(\pi +x)=\cos 2(\pi +x)+i\sin 2(\pi +x)\\ =\cos 2x+i\sin 2x\\ =f\left(x\right)\\ \therefore Periodicfunctionsarenotone-one\\ \left|f\left(x\right)\right|=\left|e^{2ix}\right|=|\cos 2x+i\sin 2x|=1\end{array}$

But in complex plane, magnitude of a complex number can vary from 0 up to infinity.

For any value of x magnitude of f(x)

Can not be greater than or less than ‘1’

f(x) does not cover the entire complex number

f is not onto.