i did this question long time ago and i realize today that at back of book answer be different.
write the following absolute value functions as piecewise functions.
g(x) = |x+1|
positive be x+1
negative be -(x+1) but in back of book it says negative one be -x-1 if x is less than -1. and i not understand.
surely you realize that -(x+1) = -x-1
|x+1| = -(x+1) if (x+1) < 0
that is, if x < -1
To write the absolute value function g(x) = |x + 1| as a piecewise function, we need to consider two cases: when x + 1 is positive and when x + 1 is negative.
1. For x + 1 > 0 (positive case):
In this case, the absolute value function is equal to the expression within the absolute value bars. Therefore, we can write g(x) = |x + 1| as g(x) = x + 1.
2. For x + 1 ≤ 0 (negative case):
Here is where the confusion lies in the back of the book. The correct expression for the negative case is -(x + 1), not -x - 1 as stated. This means that when x + 1 is negative (or zero), we need to negate the expression within the absolute value bars. Therefore, we can write g(x) = |x + 1| as g(x) = -(x + 1) if x ≤ -1.
By combining both cases, the piecewise function representation of g(x) = |x + 1| is:
g(x) = x + 1, if x > -1
g(x) = -(x + 1), if x ≤ -1