Question

What is the inverse function of `f(x) = x^2 + 9` and what is the domain and range?

Answer 1

If you restrict the domain of `f` to `x geq 0`, then `f^(-1)(x)=sqrt{x-9}` with a domain `x geq 9` and range `y geq 0`. If you restrict the domain of `f` to `x leq 0`, then `f^(-1)(x)=-sqrt{x-9}` with a domain `x geq 9` and range `y leq 0`.

Explanation:

In solving for the inverse function, you can write `y=x^2+9` and then solve for `x`: `x^2=y-9 rightarrow x= pm sqrt(y-9)`.

If you like swapping your variables, you can then do that and get the answers above based on what you want the range of the inverse function to be.