Taking Absolute Cases (1 Viewer)

[Lukybear]

When taking absolute cases, when do you know when the inequality is greater/= or just greater

I.e.

y=|x+1| - |x-3|

|x+1| = x+1 when x>= -1 etc... and why not x>-1

 

[Cazic]

You should always use greater than or equal to and less than since that is how the absolute value function is defined. That is

 

[Lukybear]

You got that straight from cambridge lol.

But how come -x is not x<= 0? Thats my actual question.

 

[jet]

because it is covered in the ≥ sign for the positive case. It only needs to be in the domain once. Though it could be ≤ and > for the domains instead.

 

[Cazic]

Well, you can do that if you want, it is equivalent in this particular case. But in general we try and avoiding "overlapping" cases, since unless we're careful it can break the well-definedness of the function, in particular, it could lead to a map being multi-valued.

For example:



Now what is f(0)?

 

[The Nomad]

Well, you can do that if you want, it is equivalent in this particular case. But in general we try and avoiding "overlapping" cases, since unless we're careful it can break the well-definedness of the function, in particular, it could lead to a map being multi-valued.

For example:



Now what is f(0)?

It's not multi-valued in the case of absolute values.

 

[Cazic]

Note my first sentence.

 

[Trebla]

I was taught it to be like this (perhaps for the sake of nice symmetry lol):


In terms of the double absolute values, it doesn't matter where you put the equal to as long as it is accounted for in one domain because of the continuity of the functions.

 

[random-1005]

You got that straight from cambridge lol.

But how come -x is not x<= 0? Thats my actual question.

because it is just convention

 

[cutemouse]

Well the definition of an absolute value is such that it is always positive.

Thus |x| = x, if x>=0 (ie. if x is positive)

and |x| = -x, if x<0 (ie. if x is negative)

You can't really say that zero is negative, hence why equality can exist in the positive case.

 

[Cazic]

What? |0| = 0 which isn't positive.

 

[cutemouse]

What? |0| = 0 which isn't positive.

Well, it isn't really negative either... if you know what I mean.

 

[Cazic]

I do know what you mean, but

Well the definition of an absolute value is such that it is always positive.

is wrong.

 

[cutemouse]

I do know what you mean, but



is wrong.

That's how I was taught it.

 

[xV1P3R]

Wasn't absolute value something like
|x| = sqrt(x²)
Or something about scalar quantity like in complex numbers, modulus

 

[Cazic]

That's how I was taught it.

Well 0 isn't a positive number, I don't know what else to tell you. You can say the absolute value function is "non-negative" though.

Wasn't absolute value something like
|x| = sqrt(x²)
Or something about scalar quantity like in complex numbers, modulus

That identity you gave only works for real numbers. In general:



which with a bit of Pythagoras you can see is indeed the "modulus", or distance from 0, of a complex number.

 

[untouchablecuz]

Wasn't absolute value something like
|x| = sqrt(x²)
Or something about scalar quantity like in complex numbers, modulus

the absolute value of a number is the numbers magnitude without regard to sign

thus:

if x ≥ 0, |x|=x
if x < 0, |x|=-x

my 2 cents (sorry if im just repeating)

 

[cutemouse]

Yep also |x|=\sqrt(x^2)