Question

Let f be a function from C (set of all complex numbers) to itself given by f(x) = x³. Write f⁻¹ (−1).

Answer 1

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