Question

If $\mathrm{f}:\mathrm{R}\to \mathrm{C}$ is defined by $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{2\mathrm{ix}}$ for x belongs to $R$, then $f$ is (where $C$ denotes the set of all complex numbers)

• A. One-one
• B. Onto
• C. One-one and onto
• D. Neither one-one nor onto

Answer 1

The correct option is D

Neither one-one nor onto

Explanation for the correct answer:

Step 1: Given information

Given, $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{2\mathrm{ix}}$

This can be written as,

$\mathrm{f}\left(\mathrm{x}\right)=\cos 2\mathrm{x}+i\sin 2\mathrm{x}$

We know that $\mathit{c}\mathit{o}\mathit{s}\mathrm{and}\mathit{s}\mathit{i}\mathit{n}$ functions are periodic.

Period of $\sin \left(Ax\right)=$$\frac{2\mathrm{\pi }}{\left|\mathrm{A}\right|}$

Period of$\sin 2x:$ $\frac{2\mathrm{\pi }}{2}=\mathrm{\pi }$

Period of $\cos \left(Ax\right)=$$\frac{2\mathrm{\pi }}{\left|\mathrm{A}\right|}$

Period of $\cos 2x:$ $\frac{2\mathrm{\pi }}{2}=\mathrm{\pi }$

$\begin{array}{rcl}\therefore \mathrm{f}(\mathrm{\pi }+\mathrm{x})& =& \cos 2(\mathrm{\pi }+\mathrm{x})+\mathrm{i}\sin 2(\mathrm{\pi }+\mathrm{x})\\ & =& \cos 2\mathrm{x}+\mathrm{i}\sin 2\mathrm{x}\\ & =& \mathrm{f}\left(\mathrm{x}\right)\end{array}$

Thus,$f\left(x\right)$is also periodic with a period of $\mathrm{\pi }$.

Since periodic functions are not one-one, $f\left(x\right)$ is not one-one.

Step 2: Finding the magnitude of $f\left(x\right):$

$$\begin{gathered} =\left|\mathrm{f}\left(\mathrm{x}\right)\right| \\ =\left|{\mathrm{e}}^{\mathrm{i}2\mathrm{x}}\right| \\ =\sqrt{{\cos }^{2}2\mathrm{x}+{\sin }^{2}2\mathrm{x}}\left[\because {\sin }^2\theta +{\cos }^2\theta =1\right] \\ =1 \end{gathered}$$

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

Since, for any value of$x$, the magnitude of $f\left(x\right)$ cannot be greater than or lesser than 1 (i.e., $f\left(x\right)$ doesn't cover the entire complex plane), we can say that $f\left(x\right)$ is not onto.

Hence, the correct answer is an option (D).