Question

Let $S$ be the set of points in the complex plane corresponding to the unit circle. That is $S=[z:|z|=1]$. Consider the function $f\left(z\right)=z{z}^{\ast }$ wherer $z^{\ast }$ denotes wth complex conjugate of $z$. The $f\left(z\right)$ maps S to which one of the following in the complex plane

• A. unit circle
• B. horizontal axis line segment from orgin to ($1,0$)
• C. the point $(1,0)$
• D. the entire horizontal axis

Answer 1

The correct option is C the point $(1,0)$

$f\left(z\right)=z{z}^{\ast }$ where $Z^{\ast }=\overline{Z}$
$s=z:|z|=1$
$f\left(z\right)=z\overline{z}=|z^2|=1=1+0i=(1,0)$
$\therefore f\left(z\right)$ maps $s$ to the $(1,0)$ in the complex plane.