The function f(x)=(x-5)^2+2 is not one to one. identify a restricted domain that makes the function one to one and find the inverse function

Answer 1

To make the function f(x) = (x-5)^2 + 2 one-to-one, we can restrict the domain.

Let's consider the original function, f(x) = (x-5)^2 + 2.

Notice that the function is a parabola with its vertex at (5,2), opening upwards. This means that for any given y-coordinate, there are two corresponding x-coordinates. Therefore, the function is not one-to-one.

To make the function one-to-one, we can restrict the domain to one side of the vertex. Let's choose the right side of the vertex (x ≥ 5). This means we will only consider x-values equal to or greater than 5.

Now, let's find the inverse function.

1. Start with the original function: f(x) = (x-5)^2 + 2.
2. Replace f(x) with y: y = (x-5)^2 + 2.
3. Swap x and y: x = (y-5)^2 + 2.
4. Solve for y:
x - 2 = (y-5)^2.
√(x - 2) = y - 5.
y = √(x - 2) + 5.

So, the inverse function is f^(-1)(x) = √(x - 2) + 5, where x is limited to values equal to or greater than 5.