Restricting the Domain to Find the Inverse of a Polynomial Function

The original function $f(x)={(x-4)}^2$ is not one-to-one, but the function is restricted to a domain of $x\ge 4$ or $x\le 4$ on which it is one-to-one. See Figure 5.

To find the inverse, start by replacing $f(x)$ with the simple variable $y.$

This is not a function as written. We need to examine the restrictions on the domain of the original function to determine the inverse. Since we reversed the roles of $x$ and $y$ for the original $f(x),$ we looked at the domain: the values $x$ could assume. When we reversed the roles of $x$ and $y,$ this gave us the values $y$ could assume. For this function, $x\ge 4,$ so for the inverse, we should have $y\ge 4,$ which is what our inverse function gives.

1. ⓐThe domain of the original function was restricted to $x\ge 4,$ so the outputs of the inverse need to be the same, $f\left(x\right)\ge 4,$ and we must use the + case:
   $$f^{-1}(x)=4+\sqrt{x}$$
2. ⓑ The domain of the original function was restricted to $x\le 4,$ so the outputs of the inverse need to be the same, $f\left(x\right)\le 4,$ and we must use the – case:
   $$f^{-1}(x)=4-\sqrt{x}$$