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

Question

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

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Answer

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

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