>> 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 even functions then
we provide 2 examples
of even functions,
and then we determine whether
each of the 4 functions here
is even or not.
All right, we're going
to be discussing
even functions.
A function F is called an even
function if whenever the
ordered pair X,
Y is on the graph of F,
so is negative X, Y. So,
if you have some X-coordinate
with some Y value,
there's also the opposite
of that X-coordinate.
So, if you have 2, 3,
you'll also have negative 2,
3 for instance.
So, this is how we define an
even function right here
in the box,
that's the definition.
A function F 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. That means when you plug
in negative X for X,
you get exactly the
same function.
The graph of an even function
is symmetric with the Y-axis.
So, here are some examples
of even functions.
F of X equals X squared
minus 4.
This first one is a parabola
going up, but let's see
if the definition makes sense.
It says that to be an even
function, F
of negative X is the
same thing.
So, let's see what negative X,
what does that mean?
That means I'm going to plug
in negative X in place of X,
so I'm putting a negative X
in place of the X here.
And then that would be X
squared minus 4.
So, notice when I computed F
of negative X,
I get exactly the
same function.
Both times,
I'm going to get X squared
minus 4, all right?
I got the same thing.
So, that shows me
that it's true that F
of negative X is actually the
same thing as F of X. Now,
what does that look
like on a graph?
So, we're just going
to sketch it quickly.
So, let's say that's 5
and negative 5.
[ Pause ]
All right, so if we're going
to sketch this, if we put in 0
for X, we get negative 4
for Y, so this is the point
on our graph.
Now, there's different ways
of getting the parabola
from here.
I like to say,
we'll this is going to look
like the graph
of Y equals X squared,
so I could go over 1 and up 1
and I get a point
over here, okay?
The other way
to do it is actually plug in 1
for X and make it a chart.
But if you plug in 1 for X,
you get 1 squared minus 4
which is negative 3,
so notice that's the point,
1 negative 3.
And then let's say
if you plugged in 2 for X,
you would get 0
because 2 squared minus 4
is 0.
The way I do it is I think
of going over 2 and at 4,
et cetera.
And if you plug in 3,
you're going to get up here
at 5, et cetera.
Well, here's the cool thing
about even function.
Every point on the right side
of the Y-axis, there's going
to be a corresponding point
on the left side.
So, you go straight
across the same distance
and that's
because if you plugged in 1,
you get the same value for Y
as if you plugged
in negative 1, right?
1 is on the right side
of the Y-axis
and negative 1 is
on the left side.
The Y value is the same.
So, instead of plugging
in all those numbers
for even functions,
it's completely symmetric
across the Y-axis.
So, if we have a point right
here, there's this
corresponding point over here,
et cetera.
Now, that's not true
of all parabolas
because if the vertex point is
not on the Y-axis,
that would not work.
So, in other words,
if you have some parabola
over here,
now that's not symmetric
across the Y-axis, okay?
All right,
what about this one?
F of X equals the absolute
value of X plus 1.
Not sure if you already have
graphed absolute value
function, so we'll go ahead
and do that now
in case it's needed here,
but first let's just take
the definition.
Is it true that F
of negative X is going
to be the same answer
as F of X?
So, the absolute value
of negative X is the same
thing as the absolute value
of X, and actually
that's true.
If you put in number like 5,
if you take the absolute value
of it, you're going
to get the same thing
as if you put an absolute
value of negative 5.
So, the absolute value
of negative X is actually the
same thing
as the absolute value
of X plus 1.
So, let's make a quick little
table of ordered pairs
for this function, X and Y,
let's say.
We'll just put in some numbers
for X and Y. So,
if you put in 0 for X
into this function,
we'll have absolute value of 0
which is 0 plus 1, okay?
Let's put in 1 for X. So,
if we put in 1 for X,
we'll get 1 plus 1 which is 2.
Now the whole point is
that if it's even,
that means I should get the
same answer
if I put a negative 1, right?
So, if I put a negative 1
for X, I'm taking the absolute
value of negative 1
which is 1, so I get 1 plus 1,
also 2, which is exactly the
definition
of an even function.
If we put in 2,
the absolute value
of 2 is 2 plus 1 or 3,
and I should get the same
answer for Y
if I put a negative 2,
which I do.
So, just in case you had never
graphed that before,
I wanted to give you a little
chart and then we're going
to graph this.
[ Pause ]
All right, so let's say 1, 2,
3 4, 5, 6, et cetera.
Okay, so let's plot
these points.
We have 0, 1, we have 1, 2,
and then I also have negative
1, 2, which again, note,
it's right across the Y-axis,
there's another point.
We have 2,
3 and we have another point
over here.
If you continue, you'll find
out you have 3, 4, negative 3,
4, et cetera,
and you actually get these
V shape.
So, these are examples
of even functions, okay?
1 is a parabola
where the vertex is
on the Y-axis
and the other is the absolute
value function 1,
an absolute value function,
where the vertex,
this point
in the center is also called
the vertex, you know, the--
of this absolute
value function.
So, both of them are
on the Y-axis.
Okay. So, those are just two
examples of even functions
and to find
out if something is an even
function, you can graph it
and look at it, and say,
"Well, if I have got 1 point
over here, is there always
in a-- a point
on the other side?"
So, if you have a point
on the left side
of the Y-axis,
is there a corresponding point
to the right straight across,
same distance.
So, in other words,
this distance
to the Y-axis has
to be the same distance,
right?
Straight over.
All right, algebraically,
you put in F of negative X
and you see
if you could make it look
exactly the same of F of X.
And so, that's what we're
going to do now.
I'm going to give you some
possibilities of functions
and then we're--
instead of graphing it,
we'll just plug in F
of negative X so we'll figure
out what that is
and we'll find
out if it's exactly the same
as the original function.
And if it is,
then we know it's an
even function.
All right,
so let's determine whether
or not each function is even
or not.
All right.
So, to do this algebraically,
we're going to see what F
of negative X equals.
So, remember
that means I replace X
with negative X. All right.
Now, what's a negative X
to the 4th power?
That's same thing as X
to the 4th
because a negative sign
to the 4th power is going
to be positive.
And-- oops,
I forgot to put the
squared here.
And then I have minus 2
and then I have a negative X,
that quantity squared,
negative X times negative X is
X squared.
Does that look exactly the
same as the original function?
Yes, it does.
So, this is an even function.
All right,
let's do the next one.
So, I'm going to plug in G
of negative X, so I've got
to plug in negative X in place
of the X. And what do I
have here?
A negative X squared will be a
positive X squared, right?
Because the negative squared
will give me a positive,
and then I forgot
to put the minus sign here.
Let me make
that a little more clear,
plugging in negative X.
And what's this?
Minus 3 times minus X,
that's going
to give you a plus X. Is
that the same
as the original function?
No, it's not because--
oops I forgot,
sorry for these mistakes,
I forgot the 3.
I have the X squared,
that's got the correct sign,
but in my original function
for G, I have a minus sign
in front of the 3X
and here I have a plus 3X.
So, these are not exactly the
same, so this one is not even.
So, here's an example
if you have it
in function notation.
But remember you could write
functions in different ways,
you could have a set
of ordered pairs.
So, let's try these.
So, determine whether
or not each function is even
or not.
All right,
so remember what it's saying?
If you have any X
in the domain,
you've also have a negative X
in the domain
with the same Y. So,
what we've got to check is
that for each value of X,
so let's check out, here's a 2
and I got a Y value of 4.
So, that means I should have
another ordered pair
with a negative 2
in the same Y value,
and there it is.
I've got negative 2
with a positive 4 as well.
So, remember if X--
let's say X, Y and negative X,
Y have to both be
in the domain.
All right?
Any time you have an X, Y,
you also are going
to have a negative X,
Y. All right, what about 0, 5?
So, we have 0, 5.
So, if I take the opposite
of that, well,
the opposite is 0, still 0,
so I still have
that ordered pair, right?
0, 5, so there's no
problem here.
Okay, look over here,
we have a negative 1, 3,
so that means I would have
to have a positive 1,
3 and here's my positive 1, 3.
So, this is an even function.
Now, if you graphed it,
you'd be able to tell.
Let's see if I can graph this
kind of quickly.
So, I've got
to make a lot little
marks here.
So, we have to 2, 4,
which is right here,
we have 0,5,
we have negative 1,3,
we have negative 2,4,
and we have 1,3.
And so hopefully,
you see not very--
I know it's kind of small.
For every point you have
to the right of the Y-axis,
there is one directly
on the opposite side,
same distance away.
Okay, so that's a way you can
visualize it as well.
Okay, what
about this other one?
So-- oh, by the way,
this is yes, it's even, right?
Okay, so let's do the
next one.
For every Y--
I mean, I'm sorry,
for every X value,
there has to be a negative X
value with the same Y. So,
here I have 4, okay,
and I've got--
that means the Y value is 5.
I should have another ordered
pair, negative 4, 5.
So, do you see another ordered
pair in negative 4, 5?
No, you don't,
there's a negative 4, 1,
totally different.
So, this one
because there is no--
since there is no negative 4,
5, it doesn't matter
about the rest of it.
This is not going to be even.
[ Pause ]
Okay. And by the way,
we do have the 3,
2 and then the negative 3, 2,
so it doesn't matter.
It means for every ordered
pair, any time you have an X,
Y, there has
to always be a negative X,
Y no matter what.
All right,
so that's what an even
function is.
We'll be going
over what odd functions are
on the next video.
[ Pause ]
Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.