Answer
$$f^{-1}(x)=x^{3/2}$$
Work Step by Step
$$y=f(x)=x^{2/3}\hspace{1cm}x\ge0$$
To find its inverse:
1) Solve for $x$ in terms of $y$:
$$y=x^{2/3}$$ $$y = \sqrt[3]{x^2}$$ $$x^2=y^3$$
Since $x\ge0$, we take here only the positive values of $x$, which means
$$x=\sqrt{y^3}=y^{3/2}$$
2) Interchange $x$ and $y$:
$$y=x^{3/2}$$
Therefore, the inverse of function $y=f(x)=x^{2/3}$, $x\ge0$ is the function $y=x^{3/2}$. In other words, $$f^{-1}(x)=x^{3/2}$$
