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) = x³. Write f^-1(-1).

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Answer

Let f^-1(-1) = x .....(1)
⇒ f(x) = -1
⇒ x³ = -1
⇒ x³ + 1 = 0
⇒ (x + 1)(x² - x + 1) = 0
[Using the identity: a³ + b³ = (a + b)(a² - ab + b²)]
$\Rightarrow\ (\text{x}+1)(\text{x}+\omega)(\text{x}+\omega^2)=0,$ 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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