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) $]$