ANSWER
[tex]\begin{gathered} f^{-1}(x)=(x+6)^2 \\ \text{Domain =}-\infty<\text{ x }<\text{ +}\infty \end{gathered}[/tex]EXPLANATION
We want to find the inverse of:
[tex]f(x)\text{ = }\sqrt[]{x}-\text{ 6}[/tex]To do that, we will make f(x) to be y and make x the subject of the function:
[tex]\begin{gathered} y\text{ = }\sqrt[]{x}\text{- 6} \\ \Rightarrow\text{ y + 6 = }\sqrt[]{x} \\ \text{Find the square of both sides:} \\ \Rightarrow x=(y+6)^2 \end{gathered}[/tex]Now, x becomes f-1(x) and y becomes x:
[tex]f^{-1}(x)=(x+6)^2[/tex]This is the inverse of the function f(x).
For the domain, we have to find the values of x such that the function can be valid.
There are no values of x that can cause the function to be invalid, so the domain is:
[tex]-\infty<\text{ x }<\text{ +}\infty[/tex]