Final answer:
The inverse function of f(x) = (x - 9)² + 3, with the domain x < 9, is f⁻¹(x) = -√(x - 3) + 9. To find this, we replaced f(x) with y, interchanged x and y, and solved for y, taking into account that we are considering the left branch of the parabola by including the negative root.
Step-by-step explanation:
The correct expression for the inverse function f⁻¹(x) of f(x) = (x - 9)² + 3 when the domain is restricted to x < 9 is f⁻¹(x) = -√(x - 3) + 9. Since the original function is a parabola opening upwards with vertex at (9,3), restricting the domain to x < 9 gives us the left branch of the parabola. The inverse of a function essentially reflects the function across the line y=x, so to find the inverse of the parabola's branch, we solve for x in terms of y.
To find the inverse, you would start by replacing f(x) with y: y = (x - 9)² + 3. Next, interchange x and y and solve for y to find the inverse:
x = (y - 9)² + 3
Subtract 3 from both sides:
x - 3 = (y - 9)²
Then take the square root of both sides, remembering to include the negative root because we are looking at x < 9:
√(x - 3) = -(y - 9)
Finally, isolate y:
y = -√(x - 3) + 9.