Question

If $R$ and $C$ denote the set of real numbers and complex numbers, respectively. Then, the function $f:C\to R$ defined by $f\left(z\right)=\left|z\right|$ is

• A. one-one
• B. onto
• C. bijective
• D. neither one-one nor onto

Answer 1

The correct option is D

neither one-one nor onto

Explanation for the correct option:

Step 1: Given information and concept

A one-one function is also known as an injective function. One element contains only one element that is a one-one correspondence
Onto function is nothing but a many-one correspondence. It is also known as the surjective function.
A bijective function is a function having both surjective as well as objective functions.

Given $f\left(z\right)=\left|z\right|$ where $z=x+iy$,

$\left|z\right|=\sqrt{x^2+y^2}$

Check the condition $f\left(z\right)=f(-z)$ if this condition satisfies then the given function is neither one-one nor onto.

$f\left(z\right)=\left|z\right|>0$,so range is not equal to co-domain.Hence, $f\left(z\right)=\left|z\right|$ is not onto.

In order to define the function, consider two complex numbers as $4+3i$ and $-4-3i$.

Step 2: Substitute the above complex numbers as shown below:

$\begin{array}{rcl}f(4+3i)& =& \left|4+3i\right|\\ & =& \sqrt{4^2+3^2}\\ & =& \sqrt{25}\\ & =& 5\end{array}$

Similarly,

$\begin{array}{rcl}f(-4-3i)& =& \left|-4-3i\right|\\ & =& \sqrt{{\left(-4\right)}^2+{\left(-3\right)}^2}\\ & =& \sqrt{16+9}\\ & =& \sqrt{25}\\ & =& 5\end{array}$

Therefore, observe that $f(4+3i)=f(-4-3i)$ then the function $f\left(z\right)$ is neither one-one nor onto.

Hence, the correct option is (D).