Expressions without absolute value signs

[garyljc]

Homework Statement

Rewrite the following expressions without absolute value signs, treating various cases separately where neccesary

Homework Equations

a-Abs[(a-(abs)a)]


the question is do i have 2 answers to this ?

 

[Hootenanny]

the question is do i have 2 answers to this ?

Yes there will be two cases. At first inspection one might conclude that there would be four cases, but you should note that one of the absolute functions is not necessary.

 

[HallsofIvy]

There is only one[/b\] function for this, which may have different parts.

If f(x)= a- |a- |x||, then, of course, you need to look at x< 0 and x> 0, then at the cases |x|> a and |x|< a.

You could also write this as a single "formula" using the Heaviside step function which is defined as H(x)= 0 if x< 0, H(x)= 1 if $x\ge 0$.

 

[garyljc]

Yes there will be two cases. At first inspection one might conclude that there would be four cases, but you should note that one of the absolute functions is not necessary.

why would there be 4 cases ?

 

[HallsofIvy]

Hmmm, the function is a-|a- |x||. If x> 0 that is a- |a- x| so if x< a, that is if a-x> 0, we have a- (a- x)= x. If x> a, that is if a- x< 0, |a-x|= -(a-x) and we have a+(a-x)= 2a- x.

If x< 0, so we have a- |a+ x|. Now if a> -x, so x+ a> 0, that is a- (a+ x)= -x. Finally if a< -x so x+ a< 0, that is a+ (a+x)= 2a+ x. Looks like 4 cases to me.