Do you know its property ?-8

Algebra Level 5

Which of the following functions are periodic ?

$\large a. ~~~\text{sgn}\left(e^{-x}\right)$

$\large b.~~~|\sin x| +\sin x$

$\large c.~~~\min(4\cos x,|x|)$

$\large d.~~~\left\lfloor x+\dfrac{1}{2}\right\rfloor +\left\lfloor x-\dfrac{1}{2}\right\rfloor +2\lfloor -x\rfloor$

Details and Assumptions:

$\lfloor\cdot\rfloor$ represents the Floor Function .
$\text{sgn}(x)$ represents the sign function. We have $\text{sgn}(x)=-1,0,1$ for negative, zero, positive $x$ respectively.

You can try more such problems of the set Do you know its property ?

a and b b and d a,b and c a,b and d

1 solution

Sanjeet Raria
Oct 16, 2014

$\Large a:$ The part under signum function is always positive throughout its domain, so result is $f(x)=1$ which is a constant function & we know that a constant function is a periodic function with no fundamental period defined.

$\Large b:$ $f(x)=2 \sin x$ when $\sin x>0$ & $f(x)= 0$ when $\sin x\leq 0$ .

So as you visualize its graph you will realize that the graph is symmetric throughout the real line. Hence periodic with period $π$

$\Large c:$ Simply by the graph in mind one can assign the non periodicity to this function.

$\Large d:$ The function can be seen repeating its value after an interval of 1, particularly at the integral points it repeats its value which is -1. So the period here is 1.

Your questions are always very nice @Sandeep Bhardwaj . This is another wisely put question.

thank you Professor Sanjeet. Your solutions are always nice. keep it up

Sandeep Bhardwaj - 6 years, 7 months ago

Are you sure that the Floor Function is periodic function??

Somdutt Goyal - 6 years, 7 months ago

Yes it is!! It repeats its value after a distance of 1 unit. Any doubt??

Sanjeet Raria - 6 years, 7 months ago

It doesn't repeat itself. In fact, the graph of Floor Function is like ascending stairs. It keeps on increasing. Isn't it?

Somdutt Goyal - 6 years, 7 months ago

@Somdutt Goyal – It appears that I've committed mistake two times. First confusing GIF with FPF when i first saw it, second- underestimating your claim.

Yes your argument is correct. I'm sorry for the mistakes. Well, silly mistakes are a part of human life but these should not left behind as it is ( otherwise that would be another big mistake). I've corrected it. Thanks for noticing the error brother @Somdutt Goyal

Sanjeet Raria - 6 years, 7 months ago

Do you know its property ?-8

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