The discussion revolves around the periodicity of the function f(x) = sin(x^2) and the implications of a change of variables in a mathematical context. Participants are exploring the definitions and characteristics of periodic functions, particularly in relation to non-linear arguments, as well as the mechanics of variable substitution in functions.
The discussion is ongoing, with participants actively questioning assumptions and definitions related to periodicity and variable changes. Some guidance has been offered regarding the exploration of zeros to identify patterns, but no consensus has been reached on the periodicity of the function in question.
Participants are grappling with the definitions of periodic functions and the implications of non-linear arguments in trigonometric functions. There is a focus on understanding the relationship between different variable representations and the nature of periodicity in mathematical functions.
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!
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?
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?