Question

If $f:R\to R$ is defined by $f\left(x\right)=x^3$ then $f^{-1}\left(8\right)=$

• A. $\left\{2\right\}$
• B. $\{2,2\omega ,2{\omega }^2\}$
• C. $\{2,-2\}$
• D. $\{2,2\omega \}$

Answer 1

Answer: (B)

$\{2,2\omega ,2{\omega }^2\}$

The inverse function of the given function is given by interchanging the $x$ and $y$ and finding the function (the inverse function which is now $y$) in terms of $x$.

For $f\left(x\right)=x^3$

Let $f\left(x\right)=y\Rightarrow x=f^{-1}\left(y\right)$

$y=x^3$

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

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

$f^{-1}\left(8\right)=8^{\frac{1}{3}}=2\times (1{)}^{\frac{1}{3}}$

We know that the cube roots of unity $\left(1\right)$ are $1,\omega$ and ${\omega }^2$

$\therefore f^{-1}\left(8\right)=\{2,2\omega ,2{\omega }^2\}$

Hence, option $B$ is correct.