Let C denote the set of all complex numbers. A function f : C → C is defined by
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)]