Question

Need help in proof
There are two functions f(x) and g(x) and two real numbers a, b. the period of the function f(x) is T1 and the period of the function g(x) is T2.
How do I prove that if T1 and T2 have common multiple, the function y = a*f(x) ± b*g(x) is periodic function and her period is equal to the lowest common multiple of T1 and T2?

Answer 1

let f and g are two functions with period Ti and To rupectively, then h" + (1+ T): f(x) and g(4+ Te) = g(x) 6t T be the least common multiple of T, and ta then To divides T - and Tz dividest a T= Tiki T= Tz kz for some k , ker 1. Now f(TC+1)= f(x+ Tiki) = f (94+ ti + Ti + ---+T,) kitimes = f (21+ + Ti + ---+]'). fel times = f (x+ 5i+tz+ ---+7) f-2 times - flutti) = f(t) a 144+1) = f60) Similarly 9(2+) = g(x) 6+ 2,5 ER Define y(a)= a f(x)+ bg ()

y latt/s af fe+) + bg(x+7) af 1600+ bg (0) s y G) T is the period of y. a