Question

If $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ is defined as $\mathrm{f}\left(\mathrm{x}\right)={(1-\mathrm{x})}^{\frac{1}{3}}$, then ${\mathrm{f}}^{-1}\left(\mathrm{x}\right)$ is equal to

• A. ${\left(1-\mathrm{x}\right)}^{\frac{-1}{3}}$
• B. ${\left(1-\mathrm{x}\right)}^3$
• C. $1-{\mathrm{x}}^3$
• D. $1-{\mathrm{x}}^{\frac{1}{3}}$

Answer 1

The correct option is C

$1-{\mathrm{x}}^3$

Explanation for the correct answer:

Given function $\begin{array}{rcl}\mathrm{f}\left(\mathrm{x}\right)& =& {\left(1-\mathrm{x}\right)}^{\frac{1}{3}}\end{array}$

Calculate the value of ${\mathrm{f}}^{-1}\left(\mathrm{x}\right)$

Let, $\begin{array}{rcl}y& =& {\left(1-\mathrm{x}\right)}^{\frac{1}{3}}\end{array}$

Take cube both sides

$$\begin{gathered} \Rightarrow {\mathrm{y}}^3=1-\mathrm{x} \\ \Rightarrow \mathrm{x}=1-{\mathrm{y}}^3 \\ \therefore {\mathrm{f}}^{-1}\left(\mathrm{x}\right)=1-{\mathrm{x}}^3 \end{gathered}$$

Hence, the correct answer is option (C).