Question

If $f:R\to R$ is given by $f\left(x\right)=x^3+3,\mathrm{then}{f}^{-1}\left(x\right)$ is equal to
(a) $x^{1/3}-3$
(b) $x^{1/3}+3$
(c) ${\left(x-3\right)}^{1/3}$
(d) $x+3^{1/3}$

Answer 1

(c)

$$\begin{gathered} \text{Let}{f}^{-1}\left(x\right)=y \\ f\left(y\right)=x \\ \Rightarrow y^3+3=x \\ \Rightarrow y^3=x-3 \\ \Rightarrow y=\sqrt[3]{x-3} \\ \Rightarrow y={\left(x-3\right)}^{\frac{1}{3}} \end{gathered}$$
So, the answer is (c).