Final answer:
The function F(x) = (x - 9)² + 3 is one-to-one, so there is no need to restrict its domain. The inverse of the function is f⁻¹(x) = √(x-3) + 9.
Step-by-step explanation:
To determine if a function is one-to-one, we need to check if it passes the horizontal line test, which means that no horizontal line intersects the graph of the function more than once. In the case of the function F(x) = (x - 9)² + 3, the graph is a parabola that opens upwards. Since every horizontal line intersects the graph at most once, the function F(x) is one-to-one.
Therefore, we do not need to restrict the domain of the function to find its inverse. The inverse of the function F(x) is found by interchanging the roles of x and y and solving for y. So, we can start with the equation x = (y - 9)² + 3 and solve for y to find the inverse.
x = (y - 9)² + 3
x - 3 = (y - 9)²
√(x-3) = y - 9
√(x-3) + 9 = y
Therefore, the inverse function of F(x) is f⁻¹(x) = √(x-3) + 9.