Question

If f : C → C is defined by f(x) = x², write f⁻¹ (−4). Here, C denotes the set of all complex numbers.

Answer 1

$$\begin{gathered} \text{Let}{f}^{-1}\left(-4\right)=x...\left(1\right) \\ \Rightarrow f\left(x\right)=-4 \\ \Rightarrow x^2=-4 \\ \Rightarrow x^2+4=0 \\ \Rightarrow \left(x+2i\right)\left(x-2i\right)=0\left[\text{u}\mathrm{sing}\text{the}\mathrm{identity}:a^2+b^2=\left(a-ib\right)\left(a+ib\right)\right] \\ \Rightarrow x=\pm 2i\left[\text{as}x\in C\right] \\ \Rightarrow f^{-1}\left(25\right)=\left\{-2i,2i\right\}\left[\mathrm{f}\text{rom}\left(1\right)\right] \end{gathered}$$