Question:
If $f: R \rightarrow R$ 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, $f^{-1}(x)=\left(\frac{x}{8}\right)^{\frac{1}{3}}$
$f^{-1}(8)=\left(\frac{8}{8}\right)^{\frac{1}{3}}$
$=1^{\frac{1}{3}}$ $(\because f: R \rightarrow R)$
Hence, if $f: R \rightarrow R$ is defined by $f(x)=8 x^{3}$ then $f^{-1}(8)=1$.
$=1$