Final answer:
To find the inverse of the restricted function F(x) = (x - 9)² + 3, the domain is restricted to x > 9 and the inverse is f⁻¹(x) = √(x - 3) + 9.
Step-by-step explanation:
To restrict the domain of the function F(x) = (x - 9)² + 3 and make it one-to-one, we need to ensure that it passes the horizontal line test, meaning it should not produce the same y-value for multiple x-values. Considering the shape of a squared function, which is a parabola, we can restrict the domain to either x > 9 or x < 9, since the vertex of this parabola is at x = 9. Let's choose the domain where x > 9. Under this restriction, the function becomes injective (one-to-one).
To find the inverse, denoted as f⁻¹(x), we solve for x in the equation y = (x - 9)² + 3 by reversing the operations that were applied to x:
- Subtract 3 from both sides to get y - 3 = (x - 9)².
- Apply the square root to both sides, noting that since x > 9, we take only the positive square root, and get √(y - 3) = x - 9.
- Add 9 to both sides, yielding f⁻¹(x) = √(y - 3) + 9, which is the correct expression for the inverse function when the domain of the original function is restricted to x > 9.
So the answer is c. restricted domain: x > 9; f⁻¹(x) = √(x - 3) + 9, though we should correct the typo and express it as f⁻¹(x) = √(x - 3) + 9.