Question:
Let $C$ denote the set of all complex numbers. A function $f: C \rightarrow C$ is defined by $f(x)=x^{3}$. Write $f^{-1}(1)$.
Solution:
Let $f^{-1}(1)=x \quad \ldots(1)$
$\Rightarrow f(x)=1$
$\Rightarrow x^{3}=1$
$\Rightarrow x^{3}-1=0$
$\Rightarrow(x-1)\left(x^{2}+x+1\right)=0 \quad\left[\right.$ Using identity : $\left.a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right]$
$\Rightarrow(x-1)(x-\omega)\left(x-\omega^{2}\right)=0$, where $\omega=\frac{1+i \sqrt{3}}{2}$
$\Rightarrow x=1, \omega$ or $\omega^{2}$ $($ as $x \in C)$
$\Rightarrow f^{-1}(1)=\left\{1, \omega, \omega^{2}\right\}$ [from (1)]