algebraically determine whether the function f(x)=|sin³x| is periodic. if it is, determine its period…

algebraically determine whether the function f(x)=|sin³x| is periodic. if it is, determine its period graphically. sketch a graph showing two periods. is the given function periodic? yes no

algebraically determine whether the function f(x)=|sin³x| is periodic. if it is, determine its period graphically. sketch a graph showing two periods. is the given function periodic? yes no

Answer

Explanation:

Step1: Recall periodic - function definition

A function (y = f(x)) is periodic if (f(x + T)=f(x)) for all (x) in the domain of (f) and some positive real - number (T). We know that the function (y = \sin x) has a period of (2\pi), i.e., (\sin(x + 2\pi)=\sin x) for all (x\in R).

Step2: Analyze (f(x)=\vert\sin^{3}x\vert)

First, consider (f(x + \pi)=\vert\sin^{3}(x+\pi)\vert). Since (\sin(x + \pi)=-\sin x), then (\sin^{3}(x + \pi)=(-\sin x)^{3}=-\sin^{3}x). And (\vert\sin^{3}(x+\pi)\vert=\vert-\sin^{3}x\vert=\vert\sin^{3}x\vert = f(x)) for all (x\in R).

Answer:

Yes