Period, period

The answer is undefined , not $2\pi$ .

Why is the answer behavior like that? Let's look at $f(3,4)=12$ . In period $12s$ , the first wave could complete four turns, and the second wave can complete three turns. It is all integer number of turns, so the answer is valid.

But if we look at $f(2,\pi)$ , and assume the answer is $2\pi$ , then in the period $2\pi$ , the first wave could complete $\pi$ turns and the second wave can complete $2$ turns. This time, only the second wave have complete integer numbers of the period but not the first, yet we couldn't find any number of $f(2,\pi)=k$ that satisfy both $\dfrac k2$ and $\dfrac k\pi$ to be integers, so the value of this function will be $\boxed{undefined}$ .

I have a video on that : ) sum of periodic functions

For the function to be periodic, an integer number of periods length 2 would have to be equal to an integer number of periods length $\pi$ . Since $\pi$ is irrational, this is not possible and the function is not periodic.

If the function r(x) were periodic then its period must be an integer multiple of the periods of p(x) and q(x) . So for some integers a and b, we would have 2a = b $\pi$ . But then $\pi$ would = 2a/b and we know $\pi$ is irrational.

None of the first four options satisfies the equation.

pi's irrational, so anything you multiply it by will never be an even integer

$LCM(a,b)$ or $GCD(a,b)$ is defined when $a$ and $b$ are precisely on the same subset of $\mathbb{C}$ i.e $LCM(42,84)$ is $84$ wheras $LCM(42,42+i84)$ is NOT DEFINED ; in a similar fashion $LCM(2,\pi)$ is not defined ; $$ To understand this concept one need to get hold of the underlying concepts of measure theory which explains the fundamental arithmetic from a different altogether approach; for ex: multiplication by k is stretching of the number line by k units etc.

For trig function y=A+sinBX, the period is given by 2pi/B. Thus we can work backwards and create functions with periods 2 and pi, namely y=pi x and y=2x. The sum of these functions is y=pi x+2x, and plugging in each of the first four options shows that none of these are the period, so the answer must be D.

The ans is option D. Simply because l.c.m. of a rational and an irrational is not possible. So r is non-periodic.