If f : C → C is defined by
Question:

If $f: C \rightarrow C$ is defined by $f(x)=8 x^{3}$, then $f^{-1}(8)=$ . _________.

Solution:

Given: $f(x)=8 x^{3}$

$f(x)=8 x^{3}$

$\Rightarrow y=8 x^{3}$

$\Rightarrow x^{3}=\frac{y}{8}$

$\Rightarrow x=\left(\frac{y}{8}\right)^{\frac{1}{3}}$

Thus, $f^{-1}(x)=\left(\frac{x}{8}\right)^{\frac{1}{3}}$

$f^{-1}(8)=\left(\frac{8}{8}\right)^{\frac{1}{3}}$

$=1^{\frac{1}{3}}$

$=1, \omega, \omega^{2} \quad(\because f: C \rightarrow C)$

where, $\omega$ is the cube root of unity.

Hence, if $f: C \rightarrow C$ is defined by $f(x)=8 x^{3}$, then $f^{-1}(8)=1, \omega, \omega^{2} .$

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