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# Intro to function symmetry

Sal explains what it means for a function to be even, odd, or neither, both algebraically and graphically. Created by Sal Khan.

Video transcript

Let's see if we can learn
a thing or two about even functions and odd functions. So even functions, and on the
right-hand side over here, we'll talk about odd functions. If we have time, we'll
talk about functions that are neither even nor odd. So before I go into kind
of a formal definition of even functions, I
just want to show you what they look like visually,
because I think that's probably the easiest way
to recognize them. And then it'll also make
a little bit more sense when we talk about the formal
definition of an even function. So let me draw some
coordinate axes here, x-axis. And then, let me
see if I can draw that a little bit straighter. This right over
here is my y-axis, or I could say y is equal to
f of x axis, just like that. And then let me draw
the graph of f of x. f of x is equal
to x-squared, or y is equal to
x-squared, either one. So let me draw it. In the first quadrant,
it looks like this. And then in the second
quadrant, it looks like this. It looks like-- oh, let me try
to draw it so it's symmetric. Pretty good job. The f of x is equal to x
squared is an even function. And the way that
you recognize it is because it has this
symmetry around the y-axis. If you take what's going
on on the right-hand side, to the right of the y-axis,
and you just reflect it over the y-axis, you get the
other side of the function. And that's what tells you
it is an even function. And I want to show you one
interesting property here. If you take any
x-value-- let's say you take a positive x-value. Let's say you take the
value, x is equal to 2. If you find f of 2,
that's going to be 4. That's going to be 4 for this
particular function for f of x is equal to x squared. 2 squared is equal to 4. And if you took the
negative version of 2-- so if you took negative 2,
and you evaluated the function there, you are also
going to get 4. And this, hopefully, or maybe
makes complete sense to you. You're like, well, Sal,
obviously if I just reflect this function
over the y-axis, that's going to be the case. Whatever function value I get at
the positive value of a number, I'm going to get
the same function value at the negative value. And this is what kind of leads
us to the formal definition. If a function is even--
or I could say a function is even if and only
of-- so it's even. And don't get confused
between the term even function and the term even number. They're completely
different kind of ideas. So there's not, at least
an obvious connection that I know of, between even
functions and even numbers or odd functions
and odd numbers. So you're an even
function if and only if, f of x is equal
to f of negative x. And the reason why I
didn't introduce this from the beginning--
because this is really the definition of
an even function-- is when you look at this, you're
like hey, what does this mean? f of x is equal to
f of negative x. And all it does mean is this. It means that if I were to
take f of 2, f of 2 is 4. So let me show you with
a particular case. f of 2 is equal to f of negative 2. And this particular case for
f of x is equal to x squared, they are both equal to 4. So really, this is
just another way of saying that the
function can be reflected, or the left side of the
function is the reflection of the right side
of the function across the vertical
axis, across the y-axis. Now just to make sure we have
a decent understanding here, let me draw a few
more even functions. And I'm going to draw some
fairly wacky things just so you really kind of learn to
visually recognize them. So let's say a
function like this, it maybe jumps up
to here, and maybe it does something like that. And then on this side,
it does the same thing. It's the reflection,
so it jumps up here, then it goes like this,
then it goes like this. I'm trying to draw it so it's
the mirror image of each other. This is an even function. You take what's going
on on the right hand side of this function and
you literally just reflect it over the y-axis, and you get the
left hand side of the function. And you could see
that even this holds. If I take some value-- let's say
that this value right here is, I don't know, 3. And let's say that f of 3 over
here is equal to, let's say, that that is 5. So this is 5. We see that f of negative 3 is
also going to be equal to 5. And that's what our definition
of an even function told us. And I can draw, let me just draw
one more to really make sure. I'll do the axis in
that same green color. Let me do one more like this. And you could have maybe some
type of trigonometric looking function that looks like
this, that looks like that. And it keeps going
in either direction. So something like this
would also be even. So all of these
are even functions. Now, you are probably thinking,
well, what is an odd function? And let me draw an
odd function for you. So let me draw the
axis once again. x-axis, y-axis, or
the f of x-axis. And to show you an
odd function, I'll give you a particular odd
function, maybe the most famous of the odd functions. This is probably the most
famous of the even functions. And it is f of x--
although there are probably other contenders for
most famous odd function. f of x is equal to x
to the third power. And it looks like-- you might
have seen the graph of it. If you haven't, you can graph
it by trying some points. It looks like that. And the way to visually
recognize an odd function is you look at what's going
on to the right of the y-axis. Once again, this is the
y-axis, this is the x-axis. You have all of this business
to the right of the y-axis. If you reflect it
over the y-axis, you would get
something like this. And if the left side of this
graph looked like this, then we would be dealing
with an even function. Clearly it doesn't. To make this an odd function, we
reflect it once over the y-axis and then reflect
it over the x-axis. Or another way to
think about it, reflect it once over the y-axis
and then make it negative. Either way, it
will get you there. Or you could even reflect
it over the x-axis and then the y-axis, so you are kind
of doing two reflections. And so clearly if
you take this up here and then you reflect it over the
x-axis, you get these values, you get this part of the
graph right over here. And if you try to do it
with a particular point, and I'm doing this to kind of
hint at what the definition, the formal definition of an
odd function is going to be. Let's try a point,
let's try 2 again. If you had the point
2, f of 2 is 8. So f of 2 is equal to 8. Now what happens if
we take negative 2? f of negative 2, negative
2 to the third power, that's just going
to be negative 8. So f of negative 2 is
equal to negative 8. And in general, if we take-- so
let me just write it over here. f of 2-- so we're just
doing one particular example from this particular function. We have f of 2 is equal
to, not f of negative 2. 8 does not equal negative 8. 8 is equal to the
negative of negative 8 because that's positive 8. So f of 2 is equal to the
negative of f of negative 2. We figured out-- just
want to make it clear-- we figured out f of 2 is 8. 2 to the third power is 8. We know that f of
negative 2 is negative 8. Negative 2 to the third
power is negative 8. So you have the
negative of negative 8, negatives cancel out,
and it works out. So in general, you
have an odd function. So here's the definition. You are dealing
with an odd function if and only if f of
x for all the x's that are defined on that
function, or for which that function is
defined, if f of x is equal to the negative
of f of negative x. Or you'll sometimes
see it the other way if you multiply both sides of
this equation by negative 1, you would get negative f of x
is equal to f of negative x. And sometimes
you'll see it where it's swapped around where
they'll say f of negative x is equal to-- let me
write that careful-- is equal to negative f of x. I just swapped these two sides. So let me just draw you
some more odd functions. So I'll do these visually. So let me draw that
a little bit cleaner. So if you have
maybe the function does something wacky like
this on the right hand side. If it was even, you
would reflect it there. But we want to have
an odd function, so we're going to
reflect it again. So the rest of the function
is going to look like this. So what I've drawn in
the non-dotted lines, this right here is
an odd function. And you could even
look at the definition. If you take some
value, a, and then you take f of a, which
would put you up here. This right here would be f of a. If you take the
negative value of that, if you took negative a
here, f of negative a is going to be down here. So f of negative
a, it's going to be the same distance from
the horizontal axis. It's not completely clear
the way I drew it just now. So it's maybe going to
be like right over here. So this right over here is going
to be f of negative a, which is the same distance from
the origin as f of a, it's just the negative. I didn't completely
draw it to scale. Let me draw one more
of these odd functions. I think you might get the point. Actually, I'll draw a very
simple odd function, just to show you that
it doesn't always have to be something crazy. So a very simple
odd function would be y is equal to x,
something like this. Whoops. y is equal going
through the origin. You reflect what's on the
right onto to the left. You get that. And then you
reflect it down, you get all of this stuff
in the third quadrant. So this is also an odd function. Now, I want to leave
you with a few things that are not odd functions
and that sometimes might be confused to be odd functions. So you might have
something like this where maybe you have a
parabola, but it's not symmetric around the y-axis. And your temptation
might be, hey, there is this symmetry
for this parabola. But it's not being
reflected around the y-axis. You don't have a situation
here where f of x is equal to f of negative x. So this is neither odd nor even. Similarly, you
might see, let's say you see a shifted
cubic function. So say you have
something like this. Let's say you have x
to the third plus 1. So f of x is equal to
x to the third plus 1, so it might look
something like this. And once again,
you will be tempted to call this an odd function. But because it's shifted up, it
is no longer an odd function. You can look at that visually. So this is f of x is equal
to x to the third plus 1. If you take what's on
the right hand side and reflect it onto
to the left hand side, you would get
something like that. And then if you were
to reflect that down, you would get
something like that. So this is not an odd function. This isn't the left
reflection and then the top-bottom
reflection of what's going on on the right hand side. This over here
actually would be.