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

Question

If $f: C \rightarrow C$ is defined by $f(x)=(x-2)^3$, write $f^{-1}(-1)$.

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Answer

Let $f ^{-1}(1)=x$. (1)
$\Rightarrow f(x)=-1$
$\Rightarrow(x-2)^3=-1$
$\Rightarrow\ \text{x}-2=-1\ \text{or }-\omega\text{ or }-\omega^2$
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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