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>> Hi, this is Julie Harland
and I'm YourMathGal.
Please visit my website
at yourmathgal.com
where you could search for any
of my videos organized
by topic.
In this video,
we define odd functions
and provide examples
of odd functions such as F
of X equals negative 2X and G
of X equals X cubed,
and then we determine whether
each of the four functions
below are odd or not.
A function F is called an odd
function if whenever the
ordered pair X,
Y is on the graph of F
so is negative X, negative Y.
So for instance,
if you have the point,
let's say negative 3, 5,
then you'd also have 3,
negative 5, right?
Both the X
and Y values would switch,
both of them would be
on the graph.
And we define it formally
in this purple box.
A function F is odd
if for every X in the domain,
negative X is also
in the domain and F
of negative X equals negative
F of X. Now,
just think about that
for a minute.
If you have F of X,
usually that's your Y value,
right?
F of X is the same thing as Y.
So what you're saying is we
put in negative X, right?
Here's that negative X,
your answer has going
to be negative Y, okay?
So F of negative X is
like negative Y. You could
think about it that way.
Also, the graph
of an odd function is
symmetric with the origin.
So we're going to look
at some examples
and then determine
if some functions are odd
or not.
So here are two examples
of odd functions
and I could show you why
they're definitely
odd functions.
So what it saying is
if you plug in-- if you--
say what F of negative X,
you should get negative F
of X. So what we're going
to do is compute F
of negative X. So in place
of X, we're going to plug
in negative X. Now,
an even function says you're
going to get exactly the same
function, but an odd function
says you're going
to get the opposite function,
so all the signs
of all the terms will
be opposites.
So when I computed F
of negative X,
notice I get 2X.
That's the opposite
of negative 2X, right?
That I could write
that if you want.
That's the same thing
as the opposite
of negative 2X, right?
That's what I just sort
of said which just the same
thing as negative F of X.
So I've just shown that F
of negative X is the same
thing as negative F
of X, right?
I'm plugging in what F of X
with just negative 2X.
Now, you don't need
to go all the way down here.
This is sort of more formal
but hope you realize that F
of negative X did end
up being the opposite of F
of X. All right, so at--
right here at this point,
it should be pretty obvious
that this is going
to be an odd function
because that is true.
Now, what if you were
to graph this,
what does this look like?
Well, that's
like Y equals negative 2X,
right?
Y equals negative 2X,
what does that look like?
[ Pause ]
So we'd have 0, 0, right?
If you put in 1 for X,
what do you get?
You get negative 2
or hopefully,
you note from the slope this
is what it's going
to look like.
You get this graph right here,
okay.
So just check
out this ordered pair
right here.
I've got the point 1,
negative 2.
That means
if I take the opposite of both
of those coordinates,
I should get another point
on the graph.
So negative 1,
positive 2 is this other
ordered pair.
And that will be true of all
of the ordered pairs
on that graph.
We say it's symmetric
across the origin.
What it means is
if you take one point
and you go through the origin
and then go the same distance,
you'll find another point
on the graph.
So if I took a point right
here, wherever it is and it go
through the origin
and then end up,
I'll get another point
over here,
whatever it happens to be.
That's what it means
to be symmetric
across the origin.
All right,
here's our next one,
G of X equals X cubed.
So first let's do
that algebraically.
What is G of negative X?
It means you're plugging
in negative X for X
and that's the same thing
as negative X cubed
and that is the opposite
of what G of X is, right?
In fact, X cubed is actually G
of X so it's the opposite of G
of X. So this again is also an
odd function.
All right,
how would we graph Y equals
X cubed?
What does that look like?
So we could just plot some
ordered pairs,
putting in some numbers for X
in solving
for Y. You see here's 3
and negative 3 and 5.
Okay, so let's just start off
with 0, 0.
Certainly, no problem there.
All right, what if I put
in 1 for X?
Or what I'm graphing is G
of X, okay?
So I'm plugging in one for X,
that gives me 1, 1.
All right, so just make a note
that if I have 1,
1 on the graph,
it should be true
that I could take the opposite
of both these ordered pairs
and get another point.
So I should also have negative
1, negative 1.
And if you actually plug
in negative 1 for X,
all right,
into this top equation?
Yes, you're going
to get negative 1 for Y.
So it is true that negative 1,
negative 1.
Now, you might think, hey,
that's a straight line, that--
no it's not.
We got to try some more
ordered pairs.
What if you put in 2?
That's tricky,
it's all the way up here
at 8 'cause 2 cubed is 8
so it's actually way up here.
I need some more space
don't I?
It's an odd function.
And so I would have to go
up a little bit higher, 5, 6,
7, 8 and we go 2,
8 and we'll also going
to have 2, negative 8
but you can't see.
What happens is you get this
graph like this, okay?
So if I have 1, 8--
I'm sorry, 2, 8 on the graph,
when you plug in negative 2,
you actually get negative 8.
But what's cool
about knowing it's an
odd function?
If you say 2, 8 is on it,
you know automatically you're
going to get negative 2,
negative 8
without actually plugging
that number
in to the actual function.
So these are examples
of two odd functions.
All right,
so let's determine whether
these functions are odd
or not.
So to do that,
we have to compute F
of negative X. So we're going
to plug in negative X for X,
and that gives you negative 3X
plus 1.
So look at the original
function, 3X plus 1
and then look at F
with negative X. Are they
the opposite?
Remember, what we're looking
for is that one is the
opposite of the other?
That's what it means
to be odd.
When they're the same,
it's even, right?
When it's exactly the same
that is an even function.
So let's look at this.
Is this the opposite?
Well, I've got a 3X
and I've got a negative 3X
but I have plus one
both times.
So this is not odd.
If this was a minus 1,
it would have worked, right?
But it's not a minus 1,
it's a plus 1.
So this function is not odd.
Okay, let's do the next one.
G of negative X. We're going
to plug in negative X for X.
[ Pause ]
So what does that give me?
I have 5 times--
all right,
when you have a negative X
cubed, you're going
to have a minus X squared,
right?
I mean, I'm sorry,
a minus X cubed.
So that's 5 times the negative
1X cubed, right?
And this is going
to be plus 4X.
Well have a negative times
a negative.
So this is a negative 5X cubed
and a plus 4X.
And so the question is,
is that the opposite
of G of X?
So look at our original
function G of X,
we have 5X cubed minus 4X.
The opposite would be negative
5X cubed plus 4X, right?
Sometimes, it's easier
for people to write
down what it's supposed
to look like.
So for this first one,
you could have said, "Well,
if that's F of X."
You might say, "Well,
for it to be negative F of X,
it would be negative 3X
minus 1."
And then when they see this
not coming up that way,
you know that's it's not going
to be odd.
But this one, well,
it's the opposite
of both signs.
In other words,
that's the same thing.
It's negative 5X cubed
minus 4X.
You don't need to write this.
At this point,
you should be able to see it
that yeah,
all the signs were changed.
So yeah, it's the opposite.
So this is going
to be an odd function
but this is just finishing out
and you're saying, "Well,
it really is minus that thing
which is G of X."
So you've just shown that G
of negative X is actually the
same thing as negative GX,
so this one is an
odd function.
[ Pause ]
All right, another way
to write a function is just a
set of ordered pairs.
So this is a function
with only four ordered pairs,
nothing else in it.
So what we found, what do we--
remember to be odd,
if it has X, Y,
it also has negative X,
negative Y, right?
Both the X and Y's have
to be opposite.
So we see, we have 4,
negative 3,
that means there should also
be a negative 4
and a positive 3,
and I see that one over here.
So that's good.
Next one, I've got a 2, 5,
so there's got
to be a negative 2
and a negative 5,
that's over here.
Now, when I checked this one,
I already have it, right?
Negative 4, 3 means over here.
I already have the 4
and negative 3.
Negative 2 and negative 5,
I already have the 2, 5.
So yes, this one is odd.
All right, next one,
if I have a 1, negative 4,
that means I also have
to have a negative 1,
positive 4, okay?
So look over here,
do you see a negative 1,
positive 4?
No, right?
There is no negative 1,
4, right?
So this is not odd.
Okay? So some functions are
even, some functions are odd,
and most functions are
neither, right?
So let's finish off
by saying writing
down the definitions for each.
So in summary,
a function is even
if for every X in the domain,
negative X is also
in the domain,
and F of negative X equals F
of X, right?
So that means,
if you have an ordered pair X,
Y, there's also the ordered
pair negative X,
Y on the graph,
and it's symmetric
with the Y axis.
A function is odd if we have--
if for every X in the domain,
negative X is also
in the domain but F
of negative X equals negative
F of X. It's the opposite
of function, right?
So even, you get the same
function, odd,
opposite of the function.
What that means,
for every ordered pair X, Y,
there's another ordered pair
negative X,
negative Y that's also
on the graph,
and it's symmetric
with the origin.
[ Pause ]
Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.