If f : R → R be defined by
Question:

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

Solution:

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

$\Rightarrow f(x)=1$

$\Rightarrow x^{4}=1$

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

$\Rightarrow\left(x^{2}-1\right)\left(x^{2}+1\right)=0$

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

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

$\Rightarrow x=\pm 1 \quad[$ as $x \in R]$

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

 

 

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