The inverse function is [tex]f^{-1} = (x + 8)^2[/tex], and the domain is the set of all real numbers equal to or larger than -8. So the correct option is C.
How to find the inverse function?
Remember that two functions are inverses if and only if:
[tex]f( f^{-1}(x)) = x\\f^{-1}(f(x)) = x[/tex]
Here we have:
[tex]f(x) = \sqrt{x} - 8[/tex]
Then we want to have:
[tex]f(f^{-1}(x)) = \sqrt{f^{-1}(x)} - 8 = x[/tex]
Solving that for the inverse function, we get:
[tex]f^{-1} = (x + 8)^2[/tex]
Now, how to get the domain. The domain of the inverse function is equal to the range of the original function.
[tex]f(x) = \sqrt{x} - 8[/tex]
The range of f(x) is the set of all real numbers larger or equal than -8 (because the smallest value that x can take is 0).
Then the domain of the inverse function is the set of all real numbers equal to or larger than -8.
The correct option is C.
If you want to learn more about inverse functions:
https://brainly.com/question/14391067
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