Mustn't Plot Them

Geometry Level 2

Which of the following functions is/are periodic?

(I) $\,\,f(x)=\left|\sin(x)\right|$
(II) $\,f(x)=\sin(\left| x \right|)$

Neither (I) nor (II) Only (I) Both (I) and (II) Only (II)

1 solution

Chew-Seong Cheong
Oct 27, 2018

Only (I) is periodic.

$f_I (x) = | \sin x |$ has only the positive half of $\sin x$ and has a period of $\pi$ that is $f_I (x) = f_I (x+\pi)$ .
$f_{II} (x) = \sin |x|$ is an even function. While $f_{II}$ is periodic with a full $sin x$ curve of $2\pi$ for $x \ge 0$ and $x\le 0$ . The curve reflects at $x=0$ where the period breaks.