If f : R → R is given by
Question:

If $f: R \rightarrow R$ is given by $f(x)=x^{3}$, write $f^{-1}$ (1)

Solution:

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

$\Rightarrow f(x)=1$

$\Rightarrow x^{3}=1$

$\Rightarrow x^{3}-1=0$

$\Rightarrow(x-1)\left(x^{2}+x+1\right)=0 \quad\left[\mathrm{u}\right.$ sing the identity $\left.: a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right]$

$\Rightarrow x=1 \quad($ as $x \in R)$

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

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