Find the Inverse f(x)=8(x+9)^(1/7)

Rewrite the equation as ${\displaystyle 8{(y+9)}^{\frac{1}{7}}=x}$.

Divide each term in ${\displaystyle 8{(y+9)}^{\frac{1}{7}}=x}$ by ${\displaystyle 8}$ and simplify.

Raise each side of the equation to the power of ${\displaystyle 7}$ to eliminate the fractional exponent on the left side.

Simplify the exponent.

Subtract ${\displaystyle 9}$ 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 ${\displaystyle f^{-1}\left(x\right)=\frac{x^7}{2097152}-9}$ is the inverse of ${\displaystyle f\left(x\right)=8{(x+9)}^{\frac{1}{7}}}$.