Which of the following function(s) is/are per | vedclass

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(D) function $f(x)$ is periodic if there exists a positive real number $T$ such that $f(x + T) = f(x)$ for all $x$ in the domain.$1$. For $f(x) = x -

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$(A)$ and $(B)$ both

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(D) function $f(x)$ is periodic if there exists a positive real number $T$ such that $f(x + T) = f(x)$ for all $x$ in the domain.$1$. For $f(x) = x - [x]$,this is the fractional part function,denoted as $\{x\}$. It is periodic with a fundamental period $T = 1$,because $\{x + 1\} = (x + 1) - [x + 1] = x + 1 - ([x] + 1) = x - [x] = \{x\}$. Thus,$(A)$ is periodic.$2$. For $w(x) = \sin^{-1} (\sin x)$,this function is periodic with a fundamental period $T = 2\pi$. Since $\sin(x + 2\pi) = \sin x$,it follows that $\sin^{-1}(\sin(x + 2\pi)) = \sin^{-1}(\sin x)$. Thus,$(B)$ is periodic.$3$. For $h(x) = x \cos x$,as $x 	o \infty$,the amplitude of the oscillations increases linearly,so it does not repeat its values at regular intervals. Thus,$(C)$ is not periodic.Therefore,both $(A)$ and $(B)$ are periodic functions.

Which of the following function$(s)$ is/are periodic?

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