Question:
If $f: C \rightarrow C$ is defined by $f(x)=x^{2}$, write $f^{-1}(-4)$. Here, $C$ denotes the set of all complex numbers.
Solution:
Let $f^{-1}(-4)=x$ $\ldots$ (1)
$\Rightarrow f(x)=-4$
$\Rightarrow x^{2}=-4$
$\Rightarrow x^{2}+4=0$
$\Rightarrow(x+2 i)(x-2 i)=0 \quad\left[\mathrm{u} \operatorname{sing}\right.$ the identity : $\left.a^{2}+b^{2}=(a-i b)(a+i b)\right]$
$\Rightarrow x=\pm 2 i$ $[$ as $x \in C]$
$\Rightarrow f^{-1}(25)=\{-2 i, 2 i\}$ $[$ from (1)]
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