Question

Let C denote the set of all complex numbers. A function f : C → C is defined by f(x) = x³. Write f⁻¹ (1).

Answer 1

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