Question:
If $f: C \rightarrow C$ is defined by $f(x)=(x-2)^{3}$, write $f^{-1}(-1)$.
Solution:
Let $f^{-1}(-1)=x$ $\ldots(1)$
$\Rightarrow f(x)=-1$
$\Rightarrow(x-2)^{3}=-1$
$\Rightarrow x-2=-1$ or $-\omega$ or $-\omega^{2}$ (as the roots of $(-1)^{\frac{1}{3}}$ are $-1,-\omega$ and $-\omega^{2}$, where $\left.\omega=\frac{1+i \sqrt{3}}{2}\right)$
$\Rightarrow x=-1+2$ or $2-\omega$ or $2-\omega^{2}=1,2-\omega, 2-\omega$
$\Rightarrow f^{-1}(-1)=\left\{1,2-\omega, 2-\omega^{2}\right\} \quad[$ from $(1)]$