Find the imagen of $R= \{z \in \mathbb{C}\mid|z-1|<1 \} $ under the tranformation $w=\frac{1}{z}$
I know that $R$ is a filled circle whit center in the point $1+0i$. But, i can't find a way to define the set for use the fact that $|w|=\frac{1}{|z|}$. I appreciate your help
Let $S=\{z\in\mathbb{C}:|z-1|<1\}$, and let $f:S\to\mathbb{C}$ be given by $f(z)={\large{\frac{1}{z}}}$.
The goal is to find $f(S)$. \begin{align*} \text{Then}\;\;&z\in f(S)\\[4pt] \iff\;&\frac{1}{z}\in S\\[4pt] \iff\;&\left|\frac{1}{z}-1\right| < 1\\[4pt] \iff\;&\left|\frac{1}{z}-1\right|^{\,2} < 1\\[4pt] \iff\;&\left(\frac{1}{z}-1\right)\left(\frac{1}{\bar{z}}-1\right) < 1\\[4pt] \iff\;&(1-z)(1-\bar{z}) < z\bar{z}\\[4pt] \iff\;&1-\bar{z}-z+z\bar{z} < z\bar{z}\\[4pt] \iff\;&z+\bar{z} > 1\\[4pt] \iff\;&\frac{z+\bar{z}}{2} > \frac{1}{2}\\[4pt] \iff\;&\text{Re}(z) > \frac{1}{2}\\[4pt] &\text{hence}\\[4pt] f(S)&=\{z\in\mathbb{C}:\text{Re}(z) > \frac{1}{2}\}\\[4pt] \end{align*} Alternatively, using the approach suggested by the user "Riemann", \begin{align*} &\left|\frac{1}{z}-1\right| < 1\\[4pt] \iff\;&|z-1| < |z|\\[4pt] \iff\;&z\;\text{is closer to}\;1\;\text{than to}\;0\\[4pt] \iff\;&\text{Re}(z) > \frac{1}{2}\\[4pt]\end{align*}