Determine whether the function is periodic. If it is periodic, find the period.
-$$f ( x ) = \left| \cos \frac { 1 } { 4 } x \right|$$
A) $4 \pi$
B) Not periodic
C) $2 \pi$
D) $\pi$

Question

Quizplus · Accepted Answer

The function $f(x) = \left| \cos \frac{1}{4}x \right|$ is periodic because the cosine function itself is periodic with a period of $2\pi$. The factor of $\frac{1}{4}$ stretches the period of the cosine function by a factor of 4, making the period of $\cos \frac{1}{4}x$ equal to $8\pi$. However, the absolute value operation does not affect the period but ensures the function remains positive. The period of the absolute value of the cosine function remains the same as the period of the cosine function itself, thus the period of $f(x)$ is $8\pi$, not $4\pi$ as initially indicated. My initial response was incorrect; the correct period for $f(x) = \left| \cos \frac{1}{4}x \right|$ is actually $8\pi$, which is not one of the provided options.

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Determine Whether the Function Is Periodic $f ( x ) = \left| \cos \frac { 1 } { 4 } x \right|$