Power Utility Function Inverse

Question

If power utility is $p = \frac{x^{1- \gamma} -1}{1 - \gamma}$ then is the inverse of the power utility function just $\frac{1 - \gamma}{p^{1- \gamma} -1}$?

Answer 1

Given that $p= \frac{x^{1-\gamma}- 1}{1-\gamma}$ we find the inverse by solving for x- Multiply both sides by $1-\gamma$: $p(1-\gamma)= x^{1-\gamma}-1$. Add 1 to both sides: $p(1-\gamma)+ 1= x^{1-\gamma}$.

Finally, take the $1-\gamma$ root of both sides: $(p(1-\gamma)+ 1)^{\frac{1}{1-\gamma}}= x$.