Question

The inverse of the function y = 9 - x 2, for all values x ≥ 3, is _____, where x can take values _____

Answer 1

F(x) = 9 - x 2 is ambiguous.  I'm going to assume that you meant

f(x) = 9 - x^2.  Use the symbol "^" to indicate exponentiation.

You are to find the inverse function.

1) replace f(x) with y.
2) interchange variables x and y:    x = 9 - y^2
3) solve the resulting equation for y:  If x = 9 - y^2, then y^2 = 9 - x, and
    there are two correct results:  y = + sqrt (9 - x) and y = - sqrt (9 - x)
4) replace "y" with "inverse function of x)."

When there are two results, as there are here, often it's best to accept the positive one.

So, let's say that the inverse of the given function is y = sqrt (9-x).
Since the domain of the sqrt function consists of zero and positive x, 9-x must be either zero or positive; thus, x: (-infinity, 9)

Check:  is x=10 OK?  No, because we'd then have sqrt(-1).
             is x=9 OK?  Yes, because we'd then have sqrt(0) = 0

And so on.