Find the Inverse 4x^(1/3)

Rewrite the equation as ${\displaystyle 4y^{\frac{1}{3}}=x}$.

Divide each term in ${\displaystyle 4y^{\frac{1}{3}}=x}$ by ${\displaystyle 4}$ and simplify.

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

Simplify the exponent.

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 ${\displaystyle f^{-1}\left(x\right)=\frac{x^3}{64}}$ is the inverse of ${\displaystyle f\left(x\right)=4x^{\frac{1}{3}}}$.