Final answer:
To find the inverse of the function F(x) = |x + 1| - 4, we express x in terms of y and consider both conditions imposed by the absolute value. The inverse function is piecewise and turns out to be F^-1(x) = x + 3 for x >= -3 and -x - 5 for x < -3.
Step-by-step explanation:
To find the inverse of the function F(x) = |x + 1| - 4, we need to express x in terms of y (or F-1(x)). Firstly, we write the equation y = |x + 1| - 4. To find the inverse, we isolate the absolute value term: y + 4 = |x + 1|. Next, we have to consider both cases due to the absolute value. For x + 1 >= 0, we get x + 1 = y + 4 and hence x = y + 3. For x + 1 < 0, we have x + 1 = -(y + 4), leading to x = -y - 5. Therefore, the inverse function is piecewise and can be written as:
F-1(x) =
{
x + 3, if x >= -3
-(x + 5), if x < -3
}
Remember that finding the inverse function often involves considering the domains where the original function is increasing or decreasing to account for the absolute value.