f(x) = −x²−9, x ≥ 0

f(f⁻¹(x)) = x ----> by definition of inverse functions

−(f⁻¹(x))²−9 = x

−(f⁻¹(x))² = x + 9

(f⁻¹(x))² = −x−9

f⁻¹(x) = √(−x−9)

or

f(x) = −x²−9, x ≤ 0

f⁻¹(x) = −√(−x−9)

The current domain is all real numbers.

The current range is y ≤ -9

We have the same value of y whenever we have corresponding negative and positive values. For example, if x = 3 we get the same value if x = -3. We need to eliminate one of the sides to get a one-to-one function. Let's arbitrarily drop the negative values.

New domain: x ≥ 0

Same range: y ≤ -9

Now to find the inverse, swap the x and y values:

x = -y² - 9

And solve in terms of y:

y² = -x - 9

y = √(-x - 9)

There's your inverse function. Notice you need to swap your domain and range too.

Answer:

y = √(-x - 9)

Domain: x ≤ -9

Range: y ≥ 0

P.S. Check the link below for the graph of the two functions. The original is red and the inverse is blue. Notice how the functions are reflected across the line y=x.

x>0

x^2 = y - 9

x = sqrt(y - 9)

y = sqrt(x - 9)