If f : C → C is defined by f(x)

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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