1. ## Damped trigonometric functions

f(x) = x sin(x). I understand that this function is to be regarded as the product of two functions (y = x and y = sin(x)), but what I don't not understand is this bit:

"Using properties of absolute and the fat that |sin x| </= 1, you have
0 </= |x||sin x| </= |x|. Consequently, -|x| </= x sin (x) </= |x|."

What is the meaning of using absolute value to prove that f(x) = x sin (x) resides within the lines y= -x and y = x? What is the justification?

Thank you, I would really like to understand this.

2. ## Re: Damped trigonometric functions

Not sure I understand your confusion.

Do you understant that as $\displaystyle \sin{x}\leq{1}$, then, for positive $\displaystyle x$,
$\displaystyle x\cdot\sin{x}\leq{x}$?

You could conceptualize it this way:
$\displaystyle x\cdot{1}=x$
So what happens when you multiply $\displaystyle x$ by a number less than 1? You get an answer that is less than $\displaystyle x$ for positive $\displaystyle x$. And as $\displaystyle |\sin{x}|\leq{1}$, then, for positive $\displaystyle x$ and positive $\displaystyle \sin{x}$, you you can conclude that:

$\displaystyle 0\leq{x\sin{x}}\leq{x}$. The absolute value signs are included because the opposite is true for negative numbers:

$\displaystyle -1\times{-2}=2$
$\displaystyle -10\times{\frac{1}{2}}=-5$
...so multiplying a negative number by a number less than $\displaystyle 1$ gives you a larger number.

This means that $\displaystyle x\sin{x}>x$ for negative $\displaystyle x$.

Or, "$\displaystyle -|x|\leq x\sin{x}$ for all $\displaystyle x$" is to say the same thing.

So, regardless of the value for $\displaystyle x$,

$\displaystyle -|x|\leq{x\sin{x}}\leq{|x|}$