Find the Inverse f(x)=5 fifth root of x+7

Rewrite the equation as ${\displaystyle 5\sqrt[5]{y+7}=x}$.

Divide each term in ${\displaystyle 5\sqrt[5]{y+7}=x}$ by ${\displaystyle 5}$ and simplify.

To remove the radical on the left side of the equation, raise both sides of the equation to the power of ${\displaystyle 5}$.

Simplify each side of the equation.

Subtract ${\displaystyle 7}$ from both sides of the equation.

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 ${\displaystyle f^{-1}\left(x\right)=\frac{x^5}{3125}-7}$ is the inverse of ${\displaystyle f\left(x\right)=5\sqrt[5]{x+7}}$.