If f : C → C is defined by f(x) = (x - 2)3, write f-1(-1). - Vidyadip

Question

If f : C → C is defined by f(x) = (x - 2)³, write f^-1(-1).

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Answer

Let f^-1(1) = x ......(1)
⇒ f(x) = -1
⇒ (x - 2)³ = -1
$\Rightarrow\ \text{x}-2=-1\ \text{or }-\omega\text{ or }-\omega^2$
as the roots of $(-1)^\frac{1}{3}$ are $-1,-\omega\text{ and }-\omega^2,$ where $\omega=\frac{1\pm\text{i}\sqrt{3}}{2}$
$\Rightarrow\ \text{x}=-1+2\text{ or }2-\omega\text{ or }2-\omega^2=1,2-\omega,2-\omega$
$\Rightarrow\ \text{f}^{-1}(-1)=1,2-\omega,2-\omega^2$ [from 1]

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