How do absolute sign works with functions?

Question

If ƒ(x)=|x| then how is:

ƒ(x)={x for x>0

-x for x<0

0 for x=0}

Since absolute sign don't take negatives into consideration should'nt x<0 be also equal to x?

Answer 1

Let's start with x = 0. If x = 0, then |x| = |0|, and |0| = 0. Since x = 0, then |x| = x

Let's now say that x > 0. In other words x is a positive number. For example, Let x = 5,

then |x| = |5| and |5| = 5, which is the same x value we started with. Each time x is a positive number, the |x| is the same positive number. If x = 6, |x| = |6| and |6| = 6, which is the same value of the x we started with.

Now, suppose the value of x is a negative number, x < 0. If x = -5, then |x| = |-5| and |-5| = 5. Since x was equal to -5, the |-5| does not equal x. Instead |-5| equals the opposite of x. The opposite of -5 is 5. Don't think of the minus sign in front of the x as a negative value. Think of it instead as the opposite of x.

If x = -6, then |x| = equals the opposite of -6 which is 6. Thus the definition of

|x| = x (the same number that is in the absolute value sign), if x = 0 or if x > 0 BUT

|x| = -x (the opposite of x), if x < 0