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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