PERIODIC FUNCTIONS

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)

a periodic function
defined for all real numbers
(B)

a periodic function
that is not defined for all real numbers
(C)

a fun periodic function

Some frequently-appearing 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:
  • whenever $\,x\,$ is in the domain of $\,f\,$, so are $\,x\pm p\,$
  • and
  • $\,f(x + p) = f(x - p) = f(x)\,$
Such a number $\,p\,$ is called a period of the function $\,f\,$.

NOTES ON THE DEFINITION:

Well-Intentioned Periodic Function Definitions that Don't Quite Work

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
the Period of a Periodic Function
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
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