Uncovering Periodicity of tan|x|: Rules & Examples

[Bassalisk]

Is tan|x| periodic and if not, why not?


I just found in my book that tan|x| isn't periodic, and how do we make up a rule how to seek periodicity of functions.

I.e.

ln(sin(x)), e^sin(x)... etc

 

[disregardthat]

Find a constant a such that f(x+a) = f(x). So basically you can solve for a function a(x) here, and make sure its a constant function.

 

[jambaugh]

Plot the graph. Notice that if you erase the coordinate axes you can still see where the origin must lie.

 

[Bassalisk]

Plot the graph. Notice that if you erase the coordinate axes you can still see where the origin must lie.

I did plot the graph, that's why i got confused. I saw a lot of functions, resembling tan(x). I got the feeling that it was periodic, but actually tan|x| isn't.

 

[jambaugh]

I did plot the graph, that's why i got confused. I saw a lot of functions, resembling tan(x). I got the feeling that it was periodic, but actually tan|x| isn't.

Look at the graph closely. It should have a mirror symmetry around x=0. But notice it is not mirror symmetric around any other point. So you can't shift the graph through a period so it looks exactly the same.

 

[Bassalisk]

Look at the graph closely. It should have a mirror symmetry around x=0. But notice it is not mirror symmetric around any other point. So you can't shift the graph through a period so it looks exactly the same.

Thank you I understand now. I had bad view of what it means to be periodic.