>> Hi, this is Julie Harland
and I'm YourMathGal.
Please visit my website
at yourmathgal.com
where you could search for any
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by topic.
We're going
to continue our work with even
and odd functions
and we'll do the following 5
problems on this video.
All right,
so here's the first problem,
go ahead and put the video
on pause and try it
on your own first.
All right, so what we need
to do is see what happens
when we plug in negative X
for X. We have to do F
of negative X. So,
we plug in negative X in place
of all the X's.
[ Pause ]
Now, remember,
if our answer ends up being F
of X, the same thing,
remember, that means it's even
and if our answer is negative
X, that means that's odd,
and if neither
of those things happens,
it's neither, so just keep
in mind-- that's a lot
of remember, when you plug
in F of negative X,
if you get the original
function, it's an even
function, and if you get
negative F of X,
the opposite, right?
All the terms will
be opposite.
If it was just 1 term,
then we would have an
odd function.
Okay, so what happens here?
Well, I've got my minus X
in the numerator,
in the denominator,
I've got a negative X cubed
and a plus 2X.
So, we have to decide,
is that the same function
or is it the opposite function
or is it neither?
And this is a little bit
tricky because when you first
look at it, you--
it doesn't look exactly
the same.
So, how about if I factor
out a negative 1
out of the numerator
and a negative 1
out of the denominator?
So, this is a tricky problem.
So, negative X, remember,
is the same thing
as negative 1, right, times X.
And if I factor
out a negative 1
out of the denominator,
I have X cubed minus 2X.
So, notice those negative 1 is
canceled and I end up with X
over X cubed minus 2X.
That's the original function,
that's exactly F of X,
so this is an even function.
Let's see,
I'm going to just write even.
[Laughs] And it's symmetric
with the Y-axis, okay?
So, we are to graph this,
so there's going
to be symmetry
with the Y-axis.
Now, this is confusing,
a lot of people look
at these fraction here
and they say, "Well,
at the top everything is
at the opposite;
at the bottom,
everything is the opposite,
so this must be an
odd function?"
Well, it's not true.
Let's just take a simple
example, take the number 8
over 1.
All right.
Now, let's just take the
opposite at the top
and the opposite
at the bottom.
That's actually exactly the
same number isn't it?
It's still 8 over 1,
it's not the opposite.
So, if you've got the opposite
of the numerator
and the opposite
of the denominator,
you're back to your original.
So, this one is worth thinking
about and looking out again,
if that isn't obvious
to you yet, okay?
So, be careful here.
This is just one term
and in one term,
you can't have the opposite
at the top and the opposite
at the bottom
and say the whole thing
is opposite.
All right,
here's another tricky one.
See if you can figure this
out on your own first.
So, we're going to do F
of negative X, so we're going
to plug in negative X for X
and we're going to plug
in negative X for X over here,
and the question is,
"Is this the same or not?"
Well, this also is
pretty tricky.
This absolute value
of negative X minus X, right?
When you simplify this,
and you know what?
This-- let's see,
this is actually the same
as this.
So, think about any number
you want.
If you take the absolute
value-- sorry,
I forgot to change my pen.
Absolute value of--
let's take 3.
It's only the same thing
as the absolute value
of negative 3.
So, that's really the same,
but then this plus X over here
and I got a minus X over here,
that makes things different.
So, it's not exactly the same
function and it's also not
exactly the opposite
of that function.
And to convince yourself
further of this,
what you could do is plug
in some numbers, right?
So, let X equal 2
for instance,
so the question is this.
"If you're going to plug in F
of 2, what do you get?"
You get the absolute value
of 2 plus 2, which is 4.
And then let's plug
in the absolute value
of negative 2.
You get negative 2 plus 2--
I'm sorry, plus negative 2.
I forgot to plug in,
I'm plugging in negative 2.
Okay, absolute value
of negative 2 is 2,
and 2 plus negative 2 is 0.
So, notice, when I plugged
in F of negative 2,
I did not get the same number
and I did not get the opposite
number, so therefore,
this is neither.
So, both of these are tricky
problems so far.
All right, next one,
try this on you own,
it's not as tricky.
Okay, so again, we're going
to plug in F of negative X.
[ Pause ]
So, we're plugging
in negative X
for every value--
every time we see an X,
we're plugging
in a negative X.
[ Pause ]
All right,
so when you square this
negative X,
we get a 3X squared.
I'm having trouble writing
today-- over X
to the 4th minus 5.
Now, this negative X squared
is also an X squared.
Is that the same function?
Yes, it is.
Nice not to have a tricky one.
So, this is an even function,
right?
And that's symmetric
with the Y-axis.
All right, the last one.
Again, put the video on pause,
try it on your own first.
So, we're going to plug
in negative X for X,
so to figure out F
of negative X. All right,
so we have a negative X
squared plus 9 times negative
X, we have a negative X
squared plus 9.
So, in the numerator,
we have X squared minus 9X.
In the denominator,
we have X squared plus 9.
So, if you look
on the numerator,
the numerator is not the same
and it's also not
the opposite.
So, without even considering
the denominator,
if the numerator isn't either
at least one or the other,
it's not going to work out,
so this also is neither.
So, this was not--
so we had a few
that were not--
were neither, right?
Neither even nor odd.
You might have noticed
if all the terms have an even
exponent then more likely
to be an even function.
For instance,
you have an X squared term
and 9, a constant is
considered
to be an even function
because you could write
that as 9X to the 0 power,
9 times 1.
But, you know what?
Threw off everything here,
X to the first power.
So, if you have a mixture
of odd and even exponents,
all right?
You got to be more careful.
Usually, the answer is going
to be neither--
I want to say usually
because there could be some
weird way the problem is set
up, maybe there's some square
roots or absolute values that,
you know, might change things
a little bit.
Right here is the
bonus problem.
So, we're going to plug
in negative X for X. So,
we have 5 times the negative X
cubed and we have 2 times the
negative X squared plus 9.
So, in the numerator,
this negative X
to the 3rd power is going
to make the whole thing
negative, right?
So, I'm just going
to write it that way.
Negative 5X cubed
over 2X squared plus 9.
So, is this the same
as F of X?
No, it's not.
Is it the opposite?
Yes, it is.
If just the numerator is the
opposite, then--
and the denominator is the
same, it's the opposite.
So, I could write this
as a negative
out in front, right?
2X squared plus 9,
this is exactly what I was
talking about, be careful.
You see an X cubed,
you also see an X squared,
you see an odd
and an even function.
But, it's the--
if one is in the numerator
and a different ones
in the denominator,
then things are different.
So, if you had a plus X
squared up here,
then you would have a problem.
So, if it's either both
in the numerator,
both in the denominator.
So, what do we have here?
Finally, this is the same
thing as negative F of X,
so this is an odd function
and it's symmetric
with the origin.
All right, so we've done lots
and lots of problems
in determining whether
functions are even or odd
and that can be very helpful
especially
when you're graphing.
[ Pause ]
Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.