Uncovering Periodicity of tan|x|: Rules & Examples

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In summary, tan|x| is not periodic because it does not have a mirror symmetry around any point other than x=0. This means that it cannot be shifted through a period and still look exactly the same.
  • #1
Bassalisk
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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
 
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  • #2
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.
 
  • #3
Plot the graph. Notice that if you erase the coordinate axes you can still see where the origin must lie.
 
  • #4
jambaugh said:
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.
 
  • #5
Bassalisk said:
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.
 
  • #6
jambaugh said:
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.
 

1. What is the period of tan|x|?

The period of tan|x| is π radians or 180 degrees. This means that the graph of tan|x| repeats itself every π radians or 180 degrees.

2. How do I determine the period of tan|x|?

To determine the period of tan|x|, you can use the formula π/|a|, where a is the coefficient of x. In the case of tan|x|, the coefficient of x is 1, so the period is π/1 = π radians or 180 degrees.

3. What are the rules for graphing tan|x|?

The rules for graphing tan|x| are the same as those for graphing a regular tangent function. The graph will have vertical asymptotes at x = nπ, where n is an integer. It will also have points of discontinuity at x = nπ/2. The graph will repeat itself every π radians or 180 degrees.

4. Can the period of tan|x| be changed?

No, the period of tan|x| cannot be changed. It is a fixed value of π radians or 180 degrees. However, you can change the amplitude of the graph by changing the coefficient of x, which will affect the steepness of the graph.

5. Can you provide an example of graphing tan|x|?

Yes, for example, the graph of y = tan|x| will have vertical asymptotes at x = nπ, where n is an integer. It will also have points of discontinuity at x = nπ/2. The graph will repeat itself every π radians or 180 degrees. See the image below for a visual representation.

Graph of tan|x|

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