## To find the inverse of the function f(x) = x^2 + 2x + 1, there are a few steps you can follow:

1. Start with the equation f(x) = y.

2. Swap the x and y variables, so that you have x = y^2 + 2y + 1.

3. Rearrange the equation to isolate y. In this case, you need to complete the square:

x = (y + 1)^2

Square root both sides of the equation:

sqrt(x) = y + 1

4. Solve for y by subtracting 1 from both sides:

sqrt(x) - 1 = y

Now that you have the equation y = sqrt(x) - 1, this represents the inverse function f^(-1)(x). The restriction x >= -1 remains, as it is a result of the original function's domain.

However, it is important to note that when finding the inverse of a function, you also need to consider the range of the original function. In this case, the original function f(x) = x^2 + 2x + 1 has a range of y >= 0, which carries over to the inverse function. Therefore, the correct final answer is:

f^(-1)(x) = sqrt(x) - 1, x >= 0

So, your final answer is indeed correct.