Final answer:
To determine if the function f(x)=(x-9)²+3 is one-to-one, we analyze its graph and find that it fails the horizontal line test. The inverse function for the restricted domain x ≥ 9 is f⁻¹(x) = √(x-3) + 9.
Step-by-step explanation:
To determine if the function f(x)=(x-9)²+3 is one-to-one, we can check its graph to see if it passes the horizontal line test. If it fails the test, it means there are two different inputs that produce the same output, and the function is not one-to-one.
By analyzing the equation (x-9)²+3, we can see that no matter what value of x we choose, the first term will always be positive. Therefore, the function will have a minimum of 3, which means it cannot be one-to-one.
To find the inverse function, we need to restrict the domain of f(x) to ensure it becomes one-to-one. The correct restricted domain is x ≥ 9. This guarantees that each output value has a unique input value.
Using this restricted domain, the inverse function is given by f⁻¹(x) = √(x-3) + 9. This function takes an input value, subtracts 3, takes the square root, and then adds 9. It will produce the original input value as the output.