To find the inverse of the function F(x) = x² + 2x over the interval [-1, 0], we need to follow these steps:
1. Replace F(x) with y to get y = x² + 2x.
2. Rearrange the equation to isolate x: y = x² + 2x becomes x² + 2x - y = 0.
3. Use the quadratic formula to solve for x in terms of y:
x = (-2 ± sqrt(4 + 4y)) / 2
x = -1 ± sqrt(1 + y)
4. Since the domain of F(x) is [-1, 0], we only consider the negative square root in the expression for x:
x = -1 - sqrt(1 + y)
5. Replace x with its inverse function notation, denoted by F^(-1)(y):
F^(-1)(y) = -1 - sqrt(1 + y)
Therefore, the inverse of the function F(x) = x² + 2x over the interval [-1, 0] is given by F^(-1)(y) = -1 - sqrt(1 + y).