Periodic function | Change of variables

In summary, f(x)=sin(x^2) is not periodic because it does not satisfy the definition of periodicity, which states that f(t+a) must be equal to f(t) for some constant a. While f(x)=sin(x) and f(x)=sin(2x) are periodic with periods of 2pi and pi, respectively, the period of f(x)=sin(x^2) is not a constant value and therefore cannot be considered periodic. Additionally, the zero set of sin(x^2) does not follow a consistent pattern, further indicating that it is not periodic.
  • #1
kingwinner
1,270
0
"Is f(x)=sin(x2) periodic?
Answer: no."

WHY? I believe "sin" is always periodic? Can someone please explain?


Any help is appreciated!
 
Last edited:
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  • #2
kingwinner said:
1) "Is f(x)=sin(x2) periodic?
Answer: no."

WHY? I believe "sin" is always periodic? Can someone please explain?



2) "Let f(x)=x.
Define the change of variable y=5x.
Then this implies g(y)=y/5.
[we have g(y)=f(x(y))=f(y/5) and f(x)=g(y(x))=g(5x)] "

If we define y=5x, then why does it imply g(y)=y/5? Shouldn't it be f(y)=y/5? Why do we need to introduce a new function g? (here we are doing a change of variable on the independent variable x, how come the dependent variable also changes?)
Also, WHY do we have g(y)=f(x(y)) and f(x)=g(y(x))? I don't understand this.


Any help is appreciated!

On the first one, what is the definition of periodic? And when you have a non-linear argument to the sin() funtion, what is the period?
 
  • #3
If x= 0, [itex]sin(x^2)= sin(0^2)= sin(0)= 0[/itex]. When is [itex]sin(x^2)= 0[/itex] again? Is that a period?
 
  • #4
berkeman said:
On the first one, what is the definition of periodic? And when you have a non-linear argument to the sin() funtion, what is the period?

I know that f(x)=sin(x) has period 2pi, g(x)=sin(2x) has period pi, etc. Since I am seeing sin in the function sin(x^2), this leads me to think that sin(x^2) is periodic as well.

For sin(x^2), The zero set is {x: x^2 = (n)(pi)}, but how can I know whether it's periodic or not?
I think it's hard for me to tell whether a function given randomly to me is periodic or not. Is there any systematic way to answer this? I am not sure where to start...

Thanks!
 
  • #5
HallsofIvy said:
If x= 0, [itex]sin(x^2)= sin(0^2)= sin(0)= 0[/itex]. When is [itex]sin(x^2)= 0[/itex] again? Is that a period?

For sin(x^2), The zero set is {x: x^2 = (n)(pi)}, but how can I know whether it's periodic or not? I just can't tell...
 
  • #6
periodic in my book would be
f(t+a) = f(t) for all t, for som constant a

in your case, solve for the first few zeros, and see what the difference bewteen them is (cf with a in the above) , see if there's any pattern which you can pick out which shows its not periodic
 

FAQ: Periodic function | Change of variables

What is a periodic function?

A periodic function is a mathematical function that repeats its values at regular intervals. In other words, it returns the same output values when the input values are increased or decreased by a certain fixed amount.

How is the period of a periodic function determined?

The period of a periodic function is the smallest positive value of x for which the function returns the same output value. It can also be determined by finding the distance between two consecutive peaks or troughs on the graph of the function.

What is the purpose of changing variables in a periodic function?

Changing variables in a periodic function can help simplify the function or make it easier to analyze. It can also help us to better understand the relationship between different variables in the function.

How is a change of variables performed in a periodic function?

A change of variables in a periodic function can be done by replacing the original variable with a new variable, such as using trigonometric functions to represent the periodicity of the function. This transformation can be done algebraically or graphically.

How does a change of variables affect the graph of a periodic function?

A change of variables can transform the graph of a periodic function in various ways. For example, it can change the amplitude, period, or phase shift of the function. It is important to carefully choose the new variable to ensure the transformed graph accurately represents the original function.

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