>> 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.
We do the following seven
problems on this video.
So remember,
here is the definition
for an EVEN function.
It's EVEN if for every X
in the domain,
negative X is also
in the domain,
and F of negative X equals F
of X. In other words,
when you replace X
with negative X,
your Y value is the same,
basically,
you get the same function.
So this means
for every ordered pair X,
Y on the graph.
The ordered pair negative X,
Y is also on the graph
and it's symmetric
with the Y-axis.
And then we've got the
definition of an ODD function.
It starts out the same,
if there's an X in the domain,
negative X must also be
in the domain.
And F of negative X though is
negative F
of X. You get the opposite
value of the function.
So this means
that for every ordered pair X,
Y on the graph,
there is another ordered pair
negative X,
negative Y that's also
on the graph.
And this means it's symmetric
with the origin.
Right, so here is our first
problem, go ahead and put it
on pause and see
if you can answer this
by yourself first.
Right, so what we want
to do is find out what F
of negative X is, right?
In every case,
to tell us something is Even
or Odd, we'll plug in F
of negative X to begin with.
So we plug in negative X
in place of X and simplify.
Oops, that's a minus sign.
Now, because a negative X
to the fourth power is just X
to fourth and a negative X
to the second power is just an
X squared.
So we've simplified this
and the question is this,
is this the same function
or is it the opposite
of the original function?
And in this case,
it's the same function, right?
So F of negative X,
if you want to finish it up,
you can say, "Oh,
that's actually equal
to F of X."
So therefore,
it's an Even function.
And that means it's symmetric
with the Y-axis.
So if you look at it,
if you look to the right
of the Y, if you graph this,
there's an ordered pair
on the right side
of the Y-axis,
there's a corresponding one
on the left side
of the X-axis,
straight across.
All right,
here is the next problem,
so go ahead and try this
on your own.
We begin the same way.
We want to figure
out what the F
of negative X is.
So we replace each X
with the negative X
and simplify.
Right, so we have a negative X
to the fifth power,
that's negative X
to the fifth.
Now, this negative X cubed is
the same thing
as negative X cubed.
And I've got a minus 4X.
So if I keep going here,
a negative times a negative
that's a plus 3X cubed
minus 4X.
All right,
so the first question,
is that exactly the same
as the original function?
No, because I have an X
to the fifth for F
of negative X.
So the second question is,
is it the opposite?
In other words,
is it negative F of X?
In other words,
are all the terms exactly the
opposite sign
of the original function?
And yes it is,
because I have an X
to the fifth here.
I've got a negative X
to the fifth here.
I have a negative 3X cubed.
I have a plus 3X cubed.
Here, I have plus 4X.
Here I have a minus 4X.
So, this is going
to be an ODD function.
So it's a symmetric
with the origin.
Now, if you want,
you can go one more step
and say, "Oh, I can write
that as the opposite of X
to the fifth minus 3X cubed
plus 4X," this is the formal
way of finishing off
the problem.
And then you've really written
that F of negative X is the
same as negative F of X.
So again, therefore,
that's why it's an odd
function and symmetric
with the origin.
All right,
here is our third problem.
F of X equals square root X.
This a little bit tricky.
So remember, to be an Even
or Odd function, it means,
if there's an X in the domain,
negative X is also
in the domain,
but with the square root
function, you can only put
positive numbers positive
numbers underneath the
square root.
So, X has to be greater
than 0, that's the domain.
Actually, it's greater than
or equal to 0, right?
Or, interval notation,
that's 0 to infinity, right?
So, if 5 is in the domain,
is negative 5 in the domain?
No, you can't graph the square
root of negative 1.
So, there's no point in trying
to plug in F of negative X
if you just think it through,
and this is Neither, okay?
So it's not Even or Odd.
If you try--
if you didn't think about it
and you just tried
to plug it in,
it gets a little bit tricky.
You have to think,
is that the same function
as this?
No, it's not 'cause I have a
minus X and also,
is it the opposite?
And actually,
it's also not the opposite.
It's a little bit tricky
because without knowing
whether X is positive
or negative, you really--
it's hard to determine
out what square root
and negative X really
means anyway.
So, in any case,
it's a tricky one
with square roots
and mainly just one
of the main points is
that the domain here,
X has to be greater than
or equal to 0.
So, if you take--
if X is positive,
then negative X is going
to be negative,
and it'll be Neither.
All right,
here is the next one.
Now, you try
to figure this one out.
Okay, so we're going to plug
in F of negative X. All right,
so we're going to plug
in negative X for X
and the question,
does that give you the
same thing?
And it doesn't.
Now, if you're not sure,
you could just square each
of these and then it'll be
more obvious.
So if I take my original
function and I square it,
it get X squared minus 6X
plus 9.
If you take negative X minus 3
and square it, this is tricky,
you get X squared, okay,
but then it's ending up going
to be plus 6X plus 9.
If you don't see that,
just go ahead and right it
out this way.
You do the FOIL method,
X squared,
but notice what happens,
negative X times negative 3 is
plus 3X, and another plus 3X,
then you have plus 9.
So that's where this
comes from.
So, notice,
these are not the same
function, right?
They're not exactly the same
because I have a minus 6X here
and a plus 6X here,
and they're also not,
you know, the opposite of one
of the other because here,
the X squared terms are the
same but the X terms are not.
So, for this function up here,
it's Neither.
You know, when it's Neither,
we don't say it's symmetric
with the Y-axis or the X-axis.
All right,
here is our next one.
Why don't you go ahead
and try that?
So we find out what F
of negative X is.
Right, by replacing X
with negative X, so,
negative X
to the third power is just
negative X cubed,
so that's negative 2X cubed
minus 7.
And so the question is,
is that exactly the same
as F of X?
No, it's not
because I have a 2X cubed
and a minus 2X cubed.
Is it the opposite?
Well, it's true
that I have a 2X cubed
and a minus 2X cubed
but I do not have a minus 7
and a plus 7,
these are the same.
So, it's Neither.
[ Pause ]
Right, here is our next one.
F of X equals X
over X squared minus one.
Let's try that.
So we're going to plug
in negative X in place of X.
So we have a negative X
in the numerator
but the denominator is an X
squared minus 1.
So what does that mean?
So the first thing,
is that the same
as the original function?
That's not true.
But is that the opposite
of it?
In other words, if you factor
out a minus sign,
you have X
over X squared minus 1,
actually, it is the opposite.
F of FX-- of F of X. So,
this is an ODD function.
So, it's symmetric
with the origin.
[ Pause ]
Right, here is our
next example.
So, feel free to put the video
on pause and try it
on your own first.
So we're going to plug
in negative X in place of X.
[ Pause ]
And then compare
to the original function.
And, that's the same
as the original function,
right?
So that means it's Even,
which means it's symmetric
with the Y-axis.
Now, the reason we're
answering this question
about whether it's symmetric
with the Y-axis is
because let's say you were
asked to graph this function.
Now, you can think of that
as Y equals 2
over X squared minus 3,
all you would have
to do is put
in positive values for X
and figure out your Ys,
and then you would just have
twice as many ordered pairs
on the left-hand side,
just putting
in the negative values for X
and keeping the same values
for Y.
[ Silence ]
Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.