If f: R R is given by f(x)=x^3, write f^ -1 (1). - Vidyadip

Question

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

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Answer

Let $f ^{-1}(1)= x$
$\Rightarrow f ( x )=1$
$\Rightarrow x^3=1$
$\Rightarrow x ^3-1=0$
$\Rightarrow(x-1)\left(x^2+x+1\right)=0\left[\right.$ Using the identity: $\left.a^3-b^3=(a-b)\left(a^2+a b+b^2\right)\right]$
$\Rightarrow x=1($ as $x \in R )$
$\Rightarrow f ^{-1}(1)=\{1\}[$ from (1)]

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