Exercise 14 Page 323

Now we solve for $y.$ Are we finished now? No. Remember that we chose the domain $x\ge \text{-}2$ for the function. When finding an inverse function the domain and range are switched. That means that the range for the inverse function must be $y\ge \text{-}2.$ Thus, we can exclude the negative radical expression.

The radical expression is only defined if $\frac{x+7}{3}$ is non-negative, which means that $x$ needs to be greater than or equal to $\text{-}7.$ This means that the domain of the inverse is $x\ge \text{-}7.$ Note that the domain and range of the inverse are reversed compared to the original function.

The $x\text{-}$values for the function are greater than $\text{-}2,$ and the $y\text{-}$values of $y^{\text{-}1}$ are also greater than $\text{-}2.$ Correspondingly, as the possible $y\text{-}$values the original function are greater than $\text{-}7,$ the same is true for the possible $x\text{-}$values of the inverse.