Let f be a function from C (set of all complex numbers) to itself given by

Question:

Let $f$ be a function from $C$ (set of all complex numbers) to itself given by $f(x)=x^{3}$. Write $f^{-1}(-1)$.

Solution:

Let $f^{-1}(-1)=x$                $\ldots(1)$

$\Rightarrow f(x)=-1$

$\Rightarrow x^{3}=-1$

$\Rightarrow x^{3}+1=0$

$\Rightarrow(x+1)\left(x^{2}-x+1\right)=0 \quad\left[\right.$ u sing the 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 $\omega=\frac{1 \pm i \sqrt{3}}{2}$

$\Rightarrow x=-1,-\omega,-\omega^{2} \quad($ as $x \in C)$

$\Rightarrow f^{-1}(-1)=\left\{-1,-\omega,-\omega^{2}\right\} \quad[$ from $(1)]$

 

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