# Periodicity of function - finding the period

#### sachinrajsharma

If $$\displaystyle f_1(x), f_2(x), f_3(x)$$ are periodic functions with period $$\displaystyle T_1,T_2,T_3$$ respectively then, we have
h(x) =$$\displaystyle af_1(x) \pm bf_2(x) \pm cf_3(x)$$ has period as,

L.C.M of $$\displaystyle (T_1,T_2,T_3)$$ ; if h(x) is not an even function

$$\displaystyle \frac{1}{2}$$L.C.M of ($$\displaystyle T_1,T_2,T_3)$$ ; if h(x) is even function

L.C.M of $$\displaystyle (\frac{a}{b},\frac{c}{d},\frac{e}{f}) \Rightarrow \frac{L.C.M of (a,c,e)}{H.C.F. of (b,d,f)}$$

Example : Find period of f(x) =tan3x +sin$$\displaystyle \frac{x}{3}$$

Period of tan3x is $$\displaystyle |\frac{\pi}{3}|$$ and period of $$\displaystyle \sin\frac{x}{3}$$ is $$\displaystyle |2\pi \times \frac{3}{1}|$$ = $$\displaystyle |6\pi|$$. Thus f(x) is periodic with period $$\displaystyle |6\pi|.$$

My question is why we have to take L.C.M and H.C.F to find the period of function. Can you guide me or give me source from where this derivation arrived?