Question

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

• A. $f$ is injective but not surjective
• B. $f$ is surjective but not injective
• C. $f$ is neither injective nor surjective
• D. $f$ is both injective and surjective

Answer 1

The correct option is C $f$ is neither injective nor surjective

Let us take 2 complex numbers

$3+4i$ and $4+3i$

$f(3+4i)=5;$ $f(4+3i)=5$.

Clearly $f\left(z\right)$ is not injective.

$f\left(z\right)$ is not surjective either as the negative real numbers are not the images of any z.