Where can I learn using absolute value function?

Question

I'm in High School and my textbooks have like skipped a lesson of modulus,i.e. $|x|$ in previous classes. I know that $$|-4|=4 $$ but how does

$$|x|=\begin{cases} x &\text{ if } x\geq0\\-x&\text{ if } x<0\end{cases}$$

This negative value of $x$ is what I don't get. If modulus gives positive values, how can $f(x)$ give a negative value?

Ps: I have found these problems while learning sets.

I went there. I got it too. Its the -x in f(x) that i dont understand — Ashish Shukla, Aug 21 '17 at 14:36
Look at the picture on the right there, with $-3$ in $f(3)$. So $|-3|=3$. — Dietrich Burde, Aug 21 '17 at 14:37
I get the |-3|=3(Sorry to sound dumb:)) Its the f(x)= |x| i dont get that says f(x) is negative when x is negative where f(x) should be just x — Ashish Shukla, Aug 21 '17 at 14:38
here is something for you https://sydney.edu.au/stuserv/documents/maths_learning_centre/absolutevalues.pdf — Dr. Sonnhard Graubner, Aug 21 '17 at 14:39
See this question and in particular the specific comment that the link leads to. — Arthur, Aug 21 '17 at 14:44

StackTD · Answer 1 · 2017-08-21T14:46:23.327

(this negative value of x is what i dont get. If modulus gives positive values,how can f(x) give a negative value)

The definition might seem confusing at first, but it is in fact very logical: $$|x| = \begin{cases} x & x \ge 0 \\ -x & x< 0\end{cases}$$ You should read this as follows:

The absolute value of $x$, written as $|x|$, is equal to:

$x$ itself if $x$ is positive;

$-x$ if $x$ is negative.

Indeed: when $x$ is negative, $-x$ is positive!

Take an example, for $x = -3$ you would get: $$|\color{red}{-3}| = -(\color{red}{-3}) = 3$$

Or put differently: we probably think of $|-3|=3$ as "dropping the minus sign", but that's hard to put into symbols if we want to write down a symbolical definition. We can however easily add an extra minus and of course this achieves the exact same thing since $-(-x) = x$.

Thank You very much......Its so clear now and you have explained it like the best way possible — Ashish Shukla, Aug 21 '17 at 14:44
@AshishShukla You're welcome and thanks; I just added a bit to my answer. — StackTD, Aug 21 '17 at 14:46

score 3 · Answer 2 · answered Aug 21 '17 at 14:41

3

Remember that $-x$ doesn't mean that the number is negative (even though there's a negative sign!) When $x$ is already negative, $-x$ is positive.

answered Aug 21 '17 at 14:41

Michael Burr

33,866

Where can I learn using absolute value function?

2 Answers2