Periodic Plus Periodic

Algebra Level 2

True or False :

Given any two periodic functions, $\color{#D61F06}{f(x)}$ and $\color{#3D99F6}{g(x)}$ , the function $\color{#69047E}{h(x)} = \color{#D61F06}{f(x)} + \color{#3D99F6}{g(x)}$ will also be periodic with a period that is, at most, the product of the original two periods (the period of $f(x)$ times the period of $g(x)$ ).

False True

3 solutions

Zandra Vinegar Staff
Nov 26, 2015

The period of $\color{#69047E}{h(x)} = \color{#D61F06}{f(x)} + \color{#3D99F6}{g(x)}$ , if it has a period, will be the Least Common Multiple (LCM) of the periods of $\color{#D61F06}{f(x)}$ and $\color{#3D99F6}{g(x)}$ whenever the LCM exists.

Many would then mistakenly claim that the LCM always exists and is, at most the product of the those two periods. However, the LCM will not exist if, for instance, one of the periods is a rational number and the other is an irrational number. In such a case, the sum of $\color{#D61F06}{f(x)}$ and $\color{#3D99F6}{g(x)}$ would be aperiodic.

Therefore, the statement is $\fbox{False}$

One interesting counter-example to the claim is that there exist two periodic functions $f,g$ whose sum $f+g$ is the identity function $x \mapsto x$ (and clearly is not periodic). This statement surprisingly requires the Axiom of Choice.

Ivan Koswara - 5 years, 6 months ago

I'm a little confused? Could you show me how my example is wrong:

f(x)=sin(x) (period 2pi) g(x)=sin(pi x) (period 2)

When I punch in h(x) = f(x) + g(x) into a graphing program it looks periodic?

Patrick Song - 5 years, 6 months ago

Initially it will look periodic. However, if you continue to follow it out either left or right, the pattern of hills and valleys slowly changes, but never repeats itself. The function is therefore not periodic.

If you’re having trouble seeing this on a traditional graphing calculator, I suggest you plug it into desmos.com.

Jason Carrier - 2 years, 10 months ago

but still they will have a LCM. Won't they ?? Root (2) * 2 = 2 * Root (2)

Vibhor Agarwal - 5 years, 6 months ago

lcm for irrational numbers is not defined!!

Kartik Arora - 5 years, 6 months ago

They will not have a LCM. Root(2)*2 is a multiple of Root(2), but it is not a multiple of 2.

Li Su - 4 years, 5 months ago

my mind over complicated it

Oximas omar - 1 month ago

This is so irritating! Clicked on True -_- No need to say what I mean

Harsh Bhimrajka SN - 5 years, 6 months ago

Pil Pinas
Jun 23, 2016

Let f(x) be a periodic function with period 1/2. Let g(x) be another periodic function with period 1/2. h(x)=f(x)+g(x) will have a period of 1/2.

However, the product of the periods of the functions f(x) and g(x) is 1/2*1/2=1/4. A contradiction to the given statement of the problem since 1/2>1/4.

The answer is FALSE.

Note that the problem emphasized on the product instead of the LCM so it won't work for fractional periods.

So far the only definitive counter example, great job.

Gabriel Kropf - 4 years, 11 months ago

Christopher Peters
Nov 29, 2015

I got this by taking the sum of two conjugate periodic functions (ex. [2x+sin(x)]+[2x-sin(x)] ) to obtain in the example a linear function (4x)

Neither of your "conjugate periodic" functions is periodic, though

Otto Bretscher - 5 years, 6 months ago

My mistake; I was thinking just [2+sin(x)]+[2-sin(x)]. But then, could a line of slope of zero be considered periodic, with an effective amplitude of zero and a period of infinity?

Christopher Peters - 5 years, 6 months ago

Yes, the constant functions are certainly periodic according to the definition that is generally used: There exists a positive $P$ such that $f(x+P)=f(x)$ for all $x$ . For the constant functions, any positive number is a period.

Otto Bretscher - 5 years, 6 months ago

It's obviously false, else it wouldn't be included in the "misconceptions" section. 😜

Chris Baldwin - 1 month ago

Periodic Plus Periodic

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