How can I prove that $f(| a|) = -|a|$? Set the function $f(x) = |x| - 2x,~x\in\mathbb{R}$. Prove that $f(|a|) = -|a|$.
How can I do this? I have no idea.
 A: You only need to use the definition: $f(|a|)=||a||-2|a|= |a|-2|a|=-|a|$. 
$||a||=|a|$ can be done, since using absolute value on a number twice equals using it only once, since after using it once, you will get a positive number, using at again will make nothing.
A: Simply plug in $x=|a|$:
$$f(|a|) = ||a||-2|a| = |a|-2|a| = -|a|. $$
Since $|a|\geq 0$, we have that $||a|| = |a|.$ This is due to the fact that the absolute value function doesn't change anything when in the input is non-negative.
A: Recall what the absolute value function does:
$f(x) = |x|$ means $f(x) = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}$, right?  Basically, if the input is positive, the absolute value is just the input.  If the input is negative, the absolute value is the positive version of the input.
Then what does $||x||$ mean?  Well, by definition, this is $= |x|$ if $|x| \geq 0$ or $-|x|$ if $|x| < 0$.  But is $|x|$ ever $< 0$?  Of course not.  So $||x|| = |x|$ for every $x$.  
In words, since $|x|$ is positive, then taking its absolute value just gives you it back since that's what the absolute value function does to positive inputs.  So $||x|| = |x|$.  Once you understand this point, the other answers should make sense to you.
