If f : C → C is defined by
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 $(1)]$

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