If f : C C is defined by f ( x )= x ^4, write f ^ -1 (1). - Vidyadip

Question

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

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Answer

Let $f ^{-1}(1)= x$ $\qquad$
$\Rightarrow f ( x )=1$
$\Rightarrow x^4=1$
$\Rightarrow x ^4-1=0$
$\Rightarrow\left(x^2-1\right)\left(x^2+1\right)=0\left[\right.$ Using identity: $\left.a^2-b^2=(a-b)(a+b)\right]$
$\Rightarrow(x-1)(x+1)(x-i)(x+i)=0$ where $i=\sqrt{-1}$ [Using identity: $\left.a^2-b^2=(a-b)(a+b)\right]$
$\Rightarrow x = \pm 1, \pm i$
$\Rightarrow f ^{-1}(1)=\{-1,1, i ,- i \}[$ from (1)]

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