If f(x) = |sin x| + |cos x| and g(x) = [x],th | vedclass

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(D) Given $f(x) = |\sin x| + |\cos x|$.We know that $f(x)$ is a periodic function with period $\frac{\pi}{2}$.Let us evaluate the range of $f(x)$:$f(x

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Periodic but period cannot be determined

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(D) Given $f(x) = |\sin x| + |\cos x|$.We know that $f(x)$ is a periodic function with period $\frac{\pi}{2}$.Let us evaluate the range of $f(x)$:$f(x)^2 = (|\sin x| + |\cos x|)^2 = \sin^2 x + \cos^2 x + 2|\sin x \cos x| = 1 + |\sin 2x|$.Since $0 \le |\sin 2x| \le 1$,we have $1 \le f(x)^2 \le 2$,which implies $1 \le f(x) \le \sqrt{2}$.Now,$h(x) = g(f(x)) = [f(x)]$.Since $1 \le f(x) \le \sqrt{2} \approx 1.414$,the value of $[f(x)]$ is always $1$ for all $x \in \mathbb{R}$.Thus,$h(x) = 1$,which is a constant function.$A$ constant function is periodic,but its fundamental period is not defined as it repeats for every real number $T > 0$.

If $f(x) = |\sin x| + |\cos x|$ and $g(x) = [x]$,then what is the period of $h(x) = g(f(x))$,where $[.]$ denotes the Greatest Integer Function $(G.I.F.)$?

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