Answer: To find the inverse function of f(x), we need to solve for x in terms of f(x). We start by writing:
y = f(x) = sqrt(x - 4) + 9
We then swap x and y and solve for y:
x = sqrt(y - 4) + 9
x - 9 = sqrt(y - 4)
(x - 9)^2 = y - 4
y = (x - 9)^2 + 4
So the inverse of f(x) is f^(-1)(x) = (x - 9)^2 + 4.
To find the domain of f^(-1)(x), we need to consider the range of f(x), which is [5, infinity). Since the inverse function swaps the roles of x and y, the domain of f^(-1)(x) is [5, infinity). So in interval notation, the domain of f^(-1)(x) is [5, infinity).
Explanation: