graph the given function f(x) over the interval -4π, 4π. determine whether the function is periodic and, if…

Explanation:

Step1: Recall periodic - function definition

A function (y = f(x)) is periodic if there exists a positive number (T) such that (f(x + T)=f(x)) for all (x) in the domain of (f). Consider the two components of (f(x)=x+\cos(3x)). The function (y = \cos(3x)) is periodic. The period of (y=\cos(ax)) is (T_1=\frac{2\pi}{|a|}), so for (y = \cos(3x)), the period (T_1=\frac{2\pi}{3}). The function (y=x) is a linear function and is non - periodic.

Step2: Analyze the sum of the two functions

Let's assume that (f(x)=x+\cos(3x)) is periodic with period (T). Then (f(x + T)=(x + T)+\cos(3(x + T))=x+\cos(3x)) for all (x). Expanding, we get (x+T+\cos(3x + 3T)=x+\cos(3x)) for all (x). This implies (T+\cos(3x + 3T)=\cos(3x)) for all (x). Since (T) is a constant and the left - hand side has a non - constant term (T) and a periodic term (\cos(3x + 3T)), and the right - hand side is just a periodic term (\cos(3x)), there is no positive real number (T) that can satisfy this equation for all (x).

Answer:

B. The function is not periodic.