Property of Period Again!

Algebra Level 4

Given that fundamental period of $g(x),h(x)$ are $A,B$ respectively. Which $A,B$ are natural numbers. And $f(x)=g(x)+h(x)$ $q(x)=g(x)×h(x)$ $1)$ Is it necessary that fundamental period of $f(x)$ is $lcm(A, B)$ ?

$2)$ Is it necessary that fundamental period of $q(x)$ is $lcm(A, B)$ ?

$•$ Fundamental period, $T$ , means the smallest number such that $f(x+T) =f(x)$ .

1) Yes, 2) No 1) No, 2) Yes 1) No, 2) No 1) Yes, 2) Yes

3 solutions

Chan Tin Ping
Feb 26, 2018

Case $1)$ : Let $g(x)=sin(\pi x),h(x)=-sin(\pi x)+5$ It is obvious that fundamental period of $g(x)=2, h(x)=2$ . However, $f(x)=5$ ,which is a constant function. Hence, the answer of question $1)$ is $\large No$ .

Case $2)$ : Let $g(x)=tan(\pi x), h(x)=2cot(\pi x)$ It is obvious that fundamental period of $g(x)=1,h(x)=1$ . However, $f(x)=2$ , which is a constant function. Hence the answer is $\large No$ .

$*$ At last, I would like to apologize because the question 'Property of Period' I posted before is $\large Wrong$ .

Note: There are actually functions where $f(x)$ has a non-zero fundamental period that is not (a multiple of) $\lcm (A,B)$ .

In the case of $g(x)$ , there is a better counterexample where $g(x)$ is non-periodic, and hence there is no fundamental period.

Calvin Lin Staff - 3 years, 3 months ago

Nivedit Jain
Mar 1, 2018

Consider an example of f(x)=sin²x and g(x)=cos²x

Hari Govind Sekhar
Feb 28, 2018

Cosider speacial case when $f(x)$ and $g(x)$ have same period $p$ .Divide the period into two halves. There might be cases in which $f(x)$ in second halve behaves as $g(x)$ in 1st halve and vise versa. In this case their product and sum have a period of $\frac{p}{2}$ .

Property of Period Again!

3 solutions

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