Question

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

Answer 1

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