Find the period of the function

[quasar987]

I have doubts concerning this problem.

Consider f:R-->R a continuous non-periodic function, and the function b(x) = cos(f(x)). Is b(x) periodic, if so, with what period?

I got...

b(x) is periodic of period L [itex]\Leftrightarrow b(x) = b(x+L) \Leftrightarrow cos(f(x)) = cos(f(x+L)) \Leftrightarrow f(x+L)=f(x)+2n\pi, \ \ n\in \mathbb{Z}[/itex]

So it depends on f wheter b is periodic of not. For exemple, if f(x) = x, then for a given n, [itex] f(x+L) = f(x)+2n\pi \Leftrightarrow x+L=x+2n\pi \Rightarrow 0<L=2n\pi[/itex] and hence the period of b(x) is [itex]2\pi[/itex].


I have faith in what I have done; it's just that it never happened to me in 21 years of life that the answer to a yes/no question in a textbook is "it depends".

 

[AKG]

You will end up with something like b being periodic iff

[tex]f(x) + 2\pi n = f(x + L) \forall x[/tex]

But then f is clearly just the function defined by:

[tex]f(x) = 2\pi nx/L + C[/tex]

for some constant C. Such a function is non-periodic (if n is not zero) and continuous. So yes, it does depend on f. If f is one of the family of functions defined by:

[tex]f(x) = 2\pi nx/L + C[/tex]

for [itex]n \in \mathbb{Z} - \{0\}[/itex], [itex]C \in \mathbb{R}[/itex], [itex]L \in \mathbb{R} - \{0\}[/itex] then the answer is yes (and with period L), and it is no otherwise. I did this in a rush, hopefully it's right.

 

[quicknote]

As a scientist, it is important to understand that not all problems have a simple yes or no answer. In this case, the periodicity of b(x) depends on the behavior of the function f(x). This is not uncommon in mathematics and science, as many problems require a deeper analysis and understanding of the underlying principles. It is important to approach problems with an open mind and be willing to explore different possibilities, even if the answer is not straightforward. Your approach to finding the period of b(x) is correct and demonstrates critical thinking and problem-solving skills. Keep in mind that science is about continuously learning and questioning, and it is perfectly normal to have doubts and uncertainties along the way.