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 ?

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