Find the Inverse x^(1/3)

Rewrite the equation as ${\displaystyle y^{\frac{1}{3}}=x}$.

Raise each side of the equation to the power of ${\displaystyle 3}$ to eliminate the fractional exponent on the left side.

Simplify the left side.

To verify the inverse, check if ${\displaystyle f^{-1}\left(f\left(x\right)\right)=x}$ and ${\displaystyle f\left(f^{-1}\left(x\right)\right)=x}$.

Evaluate ${\displaystyle f^{-1}\left(f\left(x\right)\right)}$.

Evaluate ${\displaystyle f\left(f^{-1}\left(x\right)\right)}$.

Since ${\displaystyle f^{-1}\left(f\left(x\right)\right)=x}$ and ${\displaystyle f\left(f^{-1}\left(x\right)\right)=x}$, then ${\displaystyle f^{-1}\left(x\right)=x^3}$ is the inverse of ${\displaystyle f\left(x\right)=x^{\frac{1}{3}}}$.