How do you determine a f(x) is periodic and fundamental period

[jwxie]

OKay so here are two examples:

let say
1) f(x) = cos 2x + cos x
how do you determine whether this function is periodic?
I know I have to find period for each cosine.

The first one is pi and second is 2pi. Then what do you do?
And eventually, how do you determine the fundamental period?

2) this is what I got from class but the class ended so I rushed with the note.
let say f(x) = cos 2x + 3sin (pi*x)

according to my note, this is not a periodic. How so?

and when I looked at the third examples, f(x) sin x - cos sqrt(2)*x, this is a periodic.

I just couldn't find enough guide to help me (after looking up on google).

Thank you for the help.

 

[elduderino]

for simple problems, it could be worked out common sensically. in this case, you know the individual periods of the two linear components of f(x) (namely cosx and cos2x). Now which of this would make the function return to its original value? Obviously the larger one, f(x+pi) is not equal to f(x) but f(x+2pi)=f(x).

As a general but not binding rule for such problems consider taking the least common multiple (LCM) of the involved fundamental periods.

as for the expression:f(x) = cos 2x + 3sin (pi*x)
one period is pi and the other is periodic with a fundamental period of 2. You have to remember from your notes that pi is not a rational number and hence a common multiple with an integer cannot exist. Therefore the given function is not periodic.

 

[Caesar_Rahil]

One method I found was to take individual periods of functions and then take their LCM.
Works for some, doesn't work for some
eg f(x)=tanx + cosx
period is (pi lcm 2 pi) 2*pi
but f(x) = sinx + cosx
LCM 2pi and 2pi is 2pi
but here period is pi / 2