Equation defined by function

[Dzheberloki]

Hello! I have this homework task and i cant find a proper solution to it. I found that x = 0 is a root but i wonder if there are more than just that.
Any help would be appreciated. Thanks in advance!

 

[Deleted member 4993]

Hello! I have this homework task and i cant find a proper solution to it. I found that x = 0 is a root but i wonder if there are more than just that.
Any help would be appreciated. Thanks in advance!
View attachment 22854

Did you calculate f₂(x)?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING

Please share your work/thoughts about these problems.

 

[Dzheberloki]

Did you calculate f₂(x)?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING

Please share your work/thoughts about these problems.

So I found out f₂(x) = ||x|-2|-2, f₃(x) = |||x|-2|-2|-2 ... f₂₀₂₀(x) = |||...|x|-2|-2|....-2|-2 where the number of '-2' are 2020. We know that f_n is a even function.

I also found that f₂₀₂₀(x) = { 0 , if x = 4k || -1 , if x = 2k-1 || -2 , if x = 4k+2 , where k belongs to Integer(Z) and -4040<=k<=4040
So f₂₀₂₀ belongs to [-2;0] if k is in the interval [-4040;4040]. That means the only Integer that can be root of the equation is 0. After that interval the values of f₂₀₂₀(x) are Natural numbers(N) (f₂₀₂₀(4041) = 1, f₂₀₂₀(4042) = 2 and so on /its the same for negative numbers because the function is even/).

We can also analyze the function g(x) = x/20. His values can be any number and if x>0, g(x) >0 else x<0, g(x)<0. That means there must be one more positive root which will be for x>4040 and x will probably be Real number(R) and also 2019 other negative numbers.

So I think I need to find some kind of relation between the negative roots in the form of a different function or something else. Any suggestions or help would be appreciated!