THE PERIOD OF A PERIODIC FUNCTION

Periodic functions were discussed in the previous lesson.
Be sure to read that section first!

Consider the periodic function $\,f\,$ shown above.
Assume that enough of the graph is shown to clearly illustrate the repeating part. (Nothing funny happens outside our viewing window!)
That is, the graph continues in this same way forever and ever, both to the right and to the left.

Here are two different questions (and their answers):

English lesson: the articles ‘a’ versus ‘the’

In English, the articles ‘a’ and ’the’ are used in different situations.

The article ‘the’ is used to refer to a unique object.
When you say ’the cat’, it is clear that you're referring to one particular cat.

The article ‘a’ allows for more than one object.
When you say ’a cat’, it allows for any cat, not a specific one.

So, you've got to be a bit careful with the language.
If someone stops you on the street, flashes the graph of a periodic function, and asks
‘Hey! What's the period of this function?’ they want the least positive period—the smallest one.
They don't want you to list (say) all the multiples of a number.
They just want the smallest one that works.

Let's repeat the definition of periodic function from the prior section,
this time including the special meaning of the word ‘period’ when it is preceded by the article ‘the’:

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 $\,f\,$.

When a function has a least positive period, then it is given a special name: it is called ‘the period’ of the function.

You might wonder why the definition above didn't simply say:

The least positive period is referred to as ‘‘the period’ of $\,f\,$.

Here's the problem: there are periodic functions that don't have a least positive period.
Don't worry too much—most of the periodic functions you'll deal with do indeed have a least positive period.
For a definition, though, we need to be precise.

A Periodic Function with No Least Positive Period

Consider the constant function, $\,f(x) = 7\,$.
No matter what the input is, the output is $\,7\,$.
(Any constant function would work for this example, but $\,7\,$ is the author's favorite number.)

Is $\,f\,$ periodic?
The answer is a resounding YES.
Indeed, every positive number $\,p\,$ is a period!
Why? For every real number $\,x\,$, and for every $\,p > 0\,$, we have: $$7 = f(x) = f(x + p) = f(x - p)$$ Every number in the interval $\,(0,\infty)\,$ is a period of $\,f\,$.

Is there a least positive period? In other words, is there a smallest number in the interval $\,(0,\infty)\,$?
Nope. The number zero is ‘trying to be’ the least positive period, but unfortunately $\,0\,$ isn't a positive number!

So, if you're asked ‘What is the period of $\,f(x) = 7\,$?’ you'll have to give this answer:

Sorry—constant functions don't have a least positive period.

$\bigstar$
There are also non-constant periodic functions with no least positive period.
For example, define a function $\,f\,$ to be $\,1\,$ for rational inputs, and $\,0\,$ for irrational inputs.
Then, it can be shown that every positive rational number is a period of $\,f\,$.
So, there is no least positive period.
Hint:   $\text{rational} \pm \text{rational} = \text{rational}\,$     and     $\text{irrational} \pm \text{rational} = \text{irrational}\,$
$\bigstar$ (Ray's way)
Here's another non-constant periodic function with no least positive period.
Its advantage is that it is clear, from construction, that it is indeed periodic.
Define a function $\,f\,$, as follows: it has only two outputs, $\,0\,$ and $\,1\,$.
Here's where it equals $\,1\,$:
at all the integers, as shown at right:
halfway between all those points:
halfway between all those points:
Repeat this process ad infinitum (forever and ever).
Define $\,f\,$ to equal zero everywhere else.
Clearly, $\,f\,$ is periodic! It's simply a collection of (dense) equally-spaced dots!
And, it has arbitrarily small positive periods, hence no least positive period!

The word ‘period’ can refer to a number, of a piece of the graph

To continue the potential confusion surrounding the word ‘period’, here's yet another use for the word!

Whenever you have a function $\,f\,$ with a least positive period, $\,p\,$,
then the graph of $\,f\,$ on any interval of length $\,p\,$ is often called a period or a cycle of $\,f\,$.
That is, the word ‘period’ (or ‘cycle’) is often used to refer to a smallest possible portion of the graph
that could be used as a ‘template’ to produce the entire graph.

For an example, let's return to the graph at the top of the page.

Here, the least positive period (i.e., simply ‘the period’) is $\,2\,$.

Several periods (cycles) are shown in different colors at right (green, red, purple).

Any of these pieces of graph could be used to produce the entire graph!
Just begin at its natural starting place, and keep laying it next to itself, over and over again, both to the right and to the left.

No shorter piece of graph can accomplish the same thing!

So, the word ‘period’ can refer to a number or to a piece of a graph.
How's a person to know which is wanted?
Context!!
If your teacher asks   ‘What is the period of $\,f\,$?’   then they want a (unique) number.
If your teacher says   ‘Please sketch a period of $\,f\,$’   then they want a piece of the graph.

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

When you're done practicing, move on to:
Graphs of Sine and Cosine
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7
AVAILABLE MASTERED IN PROGRESS

(MAX is 7; there are 7 different problem types.)
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