Inverse of $f(x)=x^3$ function please help me :can we inverse this function: $f(x)=x^3$? I know that if a function is a bijective function only then it can be inversed. Is this a bijective function?

Yes, indeed, it is both one-to-one (injective) and onto (surjective), and hence it is bijective. 
To show it is one-to-one, show that $f(x_1) = f(x_2) \implies x_1 = x_2$.
To show that it is onto, note that the image of $f$ is the entire codomain: the set of all real numbers. So for every $y \in \mathbb R$, there exists an $x\in \mathbb R$ such that $f(x) = y$.
So its inverse exists.

To find the inverse $f^{-1}$:


*

*Solve for $x$ as a function of $y$: $$x = \sqrt[\large 3] y$$ 

*swap $x$ for $y$: $$f^{-1}(x) = \sqrt[\large 3] x$$

