Connection between even and odd numbers and functions

Analyzing functions

When a function is positive or negative
Positive and negative parts of functions
Recognizing Odd and Even Functions
Connection between even and odd numbers and functions
Even and odd functions
Shifting functions
Shifting and reflecting functions
Recognizing features of functions (Example 1)
Recognizing features of functions 2 (example 2)
Recognizing features of functions 2 example 3)
Recognizing features of functions 2
Interpreting features of functions 2 (Example 1)
Interpreting features of functions 2 (Example 2)
Interpreting features of functions 2
Comparing features of functions 2 example 1)
Comparing features of functions 2 example 2)
Comparing features of functions 2 example 3)
Comparing features of functions 2

Connection between even and odd numbers and functions A possible reason why even functions are called "even" and odd functions are called "odd"

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Connection between even and odd numbers and functions

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In the last video on even and odd functions,
I talked about how
you shouldn't get confused between
even functions and even numbers
and odd functions and odd numbers.
And I said that there wasn't any obvious connection
between the words 'even function'
and our notion of even numbers,
or any connection between
odd functions and odd numbers.
And, I was wrong!
There actually is a relatively obvious connection
and this was pointed out by the youtube user Nothias.
And the connection,
I almost explicity did it
in the last example.
When I showed an even function,
I showed you x^2.
When I showed you an odd function,
I showed you x^3.
When I wanted to show you another odd function,
I showed you y = x, or f(x) = x^1.
And so you might start to notice
what Nothias pointed out
is that these [archetypal], or these good examples,
or these simple examples
of even and odd functions
when I just have a simple 'x' raised to some power.
Whether the power is even or odd
is going to tell you
whether the function is even or odd.
And you want to be very careful here.
Not all even or odd functions
even have exponents in them!
They could be trigonometric functions;
they might be some other type of wacky function.
You don't have to have exponents,
it's just that these exponents
are probably where the motivation
for calling these
even functions and odd functions came from.
And let me just be clear:
it's not just also any polynomial,
and even in the last video,
when we had x^3 + 1,
this was neither
even [nor] odd.
But if you just have the pure
x raised to some power,
then all of a sudden,
the motivations for calling [them] even and odd
start to make sense.
Because if I have f(x) = x^1,
that's the same thing as y = x.
This is odd!
And it kind of jells with the name
because we're also raising it
to an odd power.
If we have f(x) = x^2,
we saw in the previous video
[that] this is even.
And it kind of jells with the idea
that we're raising it to an even power.
I could keep going.
If it [were] x^3,
that is odd.
I could keep going.
Let me write it this way:
In general, if you have f(x) = x^n,
then this is odd if 'n' is odd,
and this is an even function if 'n' is even.
And I want to make it very clear here.
The whole point of this video
is to clarify the motivation
for calling them even or odd functions.
Not all even functions
are going to be of this form here,
where it's x raised to some even power,
and not all odd functions are going to be.
And I also don't want you to be confused
that if I have something like
x^3 then I have other stuff past that,
then you say," oh, x^3, that's an odd [function]!"
But this is not an odd function.
Just when it's just a pure strip-down
x^3 or x^1,
can you really make that statement.
But that really is probably
where the motivation comes
for naming them even or odd functions.
And then the other symmetric functions,
even if they don't involve an exponent,
maybe this is some type of trigonometric function,
you're calling it even because
it has the same type of symmetry
as say x^2, or x to an even power.
So you kind of group them
all together as even functions.
And then all of these,
even though this may or may not
have an exponent in it,
it has the same type of symmetries
as x raised to an odd power.
So that's why call them odd functions.

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When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...

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At 2:33, Sal says "double bonds" but should say "single bonds."

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