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### Course: Algebra (all content) > Unit 10

Lesson 36: Introduction to symmetry of functions# Function symmetry introduction

Functions can be symmetrical about the y-axis, which means that if we reflect their graph about the y-axis we will get the same graph. There are other functions that we can reflect about both the x- and y-axis and get the same graph. These are two types of symmetry we call

*even*and*odd*functions. Created by Sal Khan.## Want to join the conversation?

- i noticed that even and odd also corresponded with the exponent in all the graphs equations... coincidence? i think not!!! but seriously can anyone answer my question?(122 votes)
- In the beginning, the people saw that functions like x^2 x^4 x^6 x^8 behaved like "even functions" so they called them so. But there are other functions that behave like even functions ( cos(x) ) but don't have an even exponent.

So this is not hard and fast rule that all even functions are going to have even numbers as exponents and all functions with even numbers with exponents are going to be even functions.(73 votes)

- why is math so complicated?(16 votes)
- Math is complicated to many of us because it uses many abbreviated symbols. And math in the U.S., at least, the higher math has been taught rapidly covering far too many topics for most of us to master in such a short time. We get very little time to focus on learning, so much of our time is spent in trying to memorize what we can. Our frustration mounts when we don't have time to practice what we learn. We miss out on understanding higher concepts when we have not mastered the lower concepts. Because we are behind we often do not know how to solve problems and do not get to see the beauty of applying them to our lives. Out of desperation to make sense we give up and see math as more complicated than it ought to be. Learning takes time, practice takes more time, and mastery of math takes much longer. In other countries where math is taught slower over fewer topics the students can see how to solve problems and they get abundant practice on the topics that are presented,(96 votes)

- How can you determine whether a function is even, odd, or neither by using just the equation of the function and not by graphing it?(14 votes)
- If for all x, f(x) = f(-x) then f(x) is even.

If for all x, f(x) = -f(-x) then f(x) is odd.(34 votes)

- I don't whether my question is worth asking or not but what is the real life use of even and odd functions?(17 votes)
- They can help you with quantum mechanics.

"If a wavefunction of a particle is an even function (symmetric, centered at the origin), then the expected location of the particle (assuming it exists) is the origin, because |ψ(x)|2 is even, so x|ψ(x)|2 is odd, so E(x)≡∫x|ψ(x)|dx=0. More generally, the odd moments of an even probability distribution vanish.

A wavefunction has definite parity if it is odd or even. (It is an eigenvector with eigenvalue ±1 of the parity operator P.) One can show that generically, for a symmetric (even) potential, the energy eigenstates have parity even, odd, even, odd, ... This leads to simple numerical strategies for trying to calculate the lowest two eigenstates by restricting to even (for ground state) and odd (for lowest excited state) functions."\

For more interesting answers go here:

http://math.stackexchange.com/questions/501264/real-world-use-of-even-and-odd-functions(15 votes)

- At11:20Sal said parabola, what does that mean?(11 votes)
- A parabola is basically a line that curves in a U-shape and is symmetrical down the middle of the U (vertically). That's how I think of it anyway.(20 votes)

- So if the graph is symmetry to the y-axis, it is an even function. If the graph is symmetry to the x-axis it is an odd function?(6 votes)
- Not quite. For something to be an odd function, it has to have symmetry to the origin, not the x-axis. This means that if it has a point like (a, b), it also has the point (-a, -b). For example, y = x is an odd function because it does this.(17 votes)

- Can there be a function that is both odd and even?(6 votes)
- An odd function satisfies f(-x)= -f(x). An even function satisfies f(-x)=f(x). We want a function that does both of these. So we get -f(x)=f(x).

0=2f(x)

0=f(x)

So the only function that is both odd and even is the constant function 0.(11 votes)

- Could an even function be shaped like a V, with the point hitting the origin? Or do the functions always have to have a curved part in it(5 votes)
- Absolutely :)

y = |x|

No curves in that function!(10 votes)

- Interesting concepts and great video!

I was just wondering: is a circle centered on the origin both an even and odd function?

Ex: x^2 + y^2 = 1

Thanks!(6 votes)- A circle is
**not**a function because it fails the vertical line test. In a function, an input should have*only***one**output value.(6 votes)

- Why is it required to classify even and odd functions?(7 votes)
- i think the important thing it's to be able to understand and recognize them, which will allow you, for instance, to invert functions by restricting them at an odd interval, like with trig functions.(3 votes)

## Video transcript

- [Instructor] You've
likely heard the concept of even and odd numbers,
and what we're going to do in this video is think about
even and odd functions. And as you can see or as you will see, there's a little bit of a
parallel between the two, but there's also some differences. So let's first think about
what an even function is. One way to think about
an even function is that if you were to flip it over the y-axis, that the function looks the same. So here's a classic example
of an even function. It would be this right over here, your classic parabola where your vertex is on the y-axis. This is an even function. So this one is maybe the graph of f of x is equal to x squared. And notice, if you were to
flip it over the y-axis, you're going to get the exact same graph. Now, a way that we can talk
about that mathematically, and we've talked about this when we introduced the idea of reflection, to say that a function is equal to its reflection over the y-axis, that's just saying that f of
x is equal to f of negative x. Because if you were to replace
your x's with a negative x, that flips your function over the y-axis. Now, what about odd functions? So odd functions, you
get the same function if you flip over the y- and the x-axes. So let me draw a classic
example of an odd function. Our classic example would be f of x is equal to x to the third, is equal to x to the third, and it looks something like this. So notice, if you were to
flip first over the y-axis, you would get something
that looks like this. So I'll do it as a dotted line. If you were to flip just over the y-axis, it would look like this. And then if you were to
flip that over the x-axis, well, then you're going to
get the same function again. Now, how would we write
this down mathematically? Well, that means that our
function is equivalent to not only flipping it over the y-axis, which would be f of negative x, but then flipping that over the x-axis, which is just taking the negative of that. So this is doing two flips. So some of you might be noticing a pattern or think you might be on the
verge of seeing a pattern that connects the words even
and odd with the notions that we know from earlier
in our mathematical lives. I've just shown you an even function where the exponent is an even number, and I've just showed you an odd function where the exponent is an odd number. Now, I encourage you to try
out many, many more polynomials and try out the exponents, but it turns out that if you
just have f of x is equal to, if you just have f of x
is equal to x to the n, then this is going to be an
even function if n is even, and it's going to an odd
function if n is odd. So that's one connection. Now, some of you are thinking, "Wait, but there seem
to be a lot of functions "that are neither even nor odd." And that is indeed the case. For example, if you just had
the graph x squared plus two, this right over here is
still going to be even. 'Cause if you flip it over, you have the symmetry around the y-axis. You're going to get back to itself. But if you had x minus two squared, which looks like this, x minus two, that would
shift two to the right, it'll look like that. That is no longer even. Because notice, if you
flip it over the y-axis, you're no longer getting
the same function. So it's not just the exponent. It also matters on the structure
of the expression itself. If you have something very
simple, like just x to the n, well, then that could be or
that would be even or odd depending on what your n is. Similarly, if we were
to shift this f of x, if we were to even shift
it up, it's no longer, it is no longer, so if
this is x to the third, let's say, plus three, this is no longer odd. Because you flip it over once,
you get right over there. But then you flip it again,
you're going to get this. You're going to get something like this. So you're no longer back
to your original function. Now, an interesting thing to think about, can you imagine a function
that is both even and odd? So I encourage you to pause that video, or pause the video and
try to think about it. Is there a function where f of
x is equal to f of negative x and f of x is equal to the
negative of f of negative x? Well, I'll give you a hint, or actually I'll just give you the answer. Imagine if f of x is just
equal to the constant zero. Notice, this thing is
just a horizontal line, just like that, at y is equal to zero. And if you flip it over the y-axis, you get back to where it was before. Then if you flip it over the x-axis, again, then you're still back
to where you were before. So this over here is both even and odd, a very interesting case.