If f : C → C is defined by f(x)

Question:

If $f: C \rightarrow C$ is defined by $f(x)=x^{4}$, write $f^{-1}(1)$.

Solution:

Let $f^{-1}(1)=x \quad \ldots$ (1)

$\Rightarrow f(x)=1$

$\Rightarrow x^{4}=1$

$\Rightarrow x^{4}-1=0$

$\Rightarrow\left(x^{2}-1\right)\left(x^{2}+1\right)=0$                              [u sing identity : $a^{2}-b^{2}=(a-b)(a+b)$ ]

$\Rightarrow(x-1)(x+1)(x-i)(x+i)=0$, where $i=\sqrt{-1}$                            [u sing identity : $\left.a^{2}-b^{2}=(a-b)(a+b)\right]$          

$\Rightarrow x=\pm 1, \pm i$

$\Rightarrow f^{-1}(1)=\{-1,1, i,-i\} \quad[$ from (1) $]$          

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