Periodic functions exhibit repetitive behavior—as you move to the left or right on the graph,
there is some ‘template’ that repeats itself, forever and ever.
This section gives a precise
introduction to periodic functions.
a periodic function defined for all real numbers 
a periodic function that is not defined for all real numbers 
a fun periodic function 
Some frequentlyappearing definitions of periodic functions are a bit
‘flawed’.
The definition that follows corrects the situation.
See below for details.
DEFINITION
periodic function; period
A function $\,f\,$ is periodic
if and only if
there exists a positive number $\,p\,$ with the following properties:


$x\in\text{dom}(f)$  $\Rightarrow$  $\color{red}{x+p}\in\text{dom}(f)$ 
$x+p\in\text{dom}(f)$  $\Rightarrow$  $(x+p)+p = \color{red}{x + 2p}\in\text{dom}(f)$ 
$x+2p\in\text{dom}(f)$  $\Rightarrow$  $(x+2p)+p = \color{red}{x + 3p}\in\text{dom}(f)$ 



According to this ‘onesided’ definition, the function at right would be called periodic. (Assume that the pattern indicated extends infinitely to the right, but not to the left.) However, functions that people conventionally call ‘periodic’ repeat forever and ever, as you move to the right and as you move to the left. The definition given in this lesson remedies this situation. 
Onesided periodicity??? 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
