Let f be a function from C (set of all complex numbers) to itself… - Vidyadip

Question

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

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Answer

Let $f^{-1}(-1) = x .....(1)$
$\Rightarrow f(x) = -1$
$\Rightarrow x^3 = -1$
$\Rightarrow x^3 + 1 = 0$
$\Rightarrow (x + 1)(x^2 - x + 1) = 0$
$[$Using the identity: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)]$
$\Rightarrow\ (\text{x}+1)(\text{x}+\omega)(\text{x}+\omega^2)=0,$ where $\omega=\frac{1\pm\text{i}\sqrt{3}}{2}$
$\Rightarrow\ \text{x}=-1,-\omega,-\omega^2$ $(\text{as x}\in\text{C})$
$\Rightarrow\ \text{f}^{-1}(-1)=\{-1,-\omega,-\omega^2\} [$from $1]$

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