Hello,
Let us choose an element x of the Real Numbers. So x can either be positive or negative. So we have two case to study :
Case 1 : x is positive.
If x is positive, that is, x = 0, 1, 2, 3, ....
then the absolute value gives us the same number : |0| = 0
|1| = 1
|2|= 2
etc.
so the absolute value of x positive equals x : |x| = x
Case 2 : x is negative.
If x is negative, that is, x = - 1, - 2, -3, .....
then the absolue value gives us the opposite value : |-1| = - (-1) = 1
|-2| = -(-2) = 2
|-3| = -(-3) = 3
(Remember the product of two negative sign is a positive signe : -1 * -1 = +1)
And so we can join both cases as a piecewise function, writing :
|x| = [tex]\left \{ {{x=x if x\geq 0 } \atop {x= - x if x<0 }} \right.[/tex]
so x = x if x ≥ 0 and x = - x if x < 0
Finally the answer is : it depend on the function itself. If the function takes both positive and negative value, then the absolute value of the function will be written as a piecewise function. If the function takes only positive or negative value, then it won't be written as a piecewise function.
