[tex] \text{Consider the function}\\ \\ f(x)=6x^3-3\\ \\ \text{let }y=f(x), \text{ so we have}\\ \\ y=6x^3-3\\ \\ \text{now in order to find the inverse of the function, first we interchange x and y}\\ \text{so we get}\\ \\ x=6y^3-3 [/tex]
[tex] \text{now we will solve for y again and that will give the required inverse function.}\\ \text{so first we add 3 both sides to get}\\ \\ x+3=6y^3\\ \\ \text{now divide both sides by 6 to get}\\ \\ \frac{x+3}{6}=y^3\\ \\ \text{now take cube root both sides to get}\\ \\ \sqrt[3]{\frac{x+3}{6}}=\sqrt[3]{y^3} [/tex]
[tex] \Rightarrow \sqrt[3]{\frac{x+3}{6}}=y\\ \\ \Rightarrow y=\sqrt[3]{\frac{x+3}{6}}\\ \\ \text{hence the inverse of the function is: }\\ \\ f^{-1}(x)=\sqrt[3]{\frac{x+3}{6}} [/tex]
