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.

@khanacademy your an absolute hero when it comes to solving my math problems, I really appreciate all the effort you put into your videos, thanks a ton.

how can you tell if its a odd or even or neither by just looking at the function rather than drawing it out every time?

great video, but i think the most common Odd function would actually be f(x) = x

You just solve it algebraically; plug in "-x" into the equation. After plugging in "-x" into the equation, if the same exact equation is returned, such as in f(x) = x^2, then the equation is even. Otherwise, if the equation that is returned is the original * -1, then it is odd. For instance, f(x) = x would be an odd function.

he has a good voice.

I have to teach this concept tomorrow; I'm so nervous…

great job broo….used ur vid for my signals & systems class

keep it up dude

Great lecture, I wish my Professor was as proficient in his lectures. Thanks!

I think I missed it then. Sorry.

If only every teacher was as clear and understanding as KhanAcademy.

4:30 looks like a face lol

Brilliant as usual, thank you.

hour long lesson in 12 minutes. brilliant.

Thank you!

Oh oh oh, but there IS a relationship between even and odd numbers and functions.

f(x) = x^2 = even; 2, the exponent of x, is even.

f(x) = x^3 = odd; 3, the exponent of x, is odd.

Thank you Guruji

My teacher should just let us watch khanacdemy on our smartphones it would save everyone timee!

lmao!

…wow…

my last 2 maths teachers couldn't just say this?

"Even functions are symmetrical across the y-axis. Odd functions are symmetrical across both axis' at the same time"

Instead, they went over all these stupid calculations for about 3 hours.

Unbelievable

no, if its odd its symmetric across the origin not "both axis"

thanks man u do to me a great thing

Thanks Khan, you're the man!

The relationship between odd and even function are the exponents…. and x^2 is more likely even and x^19 is most likely to be odd

The second even function you drew looks like an angry face XD

Every time I watch these I admire your mouse control. I would never be able to write words so neatly with a mouse..

Also, your concept explanation is impeccable – the way you introduce new concepts is very understandable and easy to comprehend, I just wish i didn't leave all my study to the last minute so I could watch all your videos!

Same here. I feel ya

You sir are the reason I pass classes

you're so awesome =D

literally the only thing i say in school when i dont get something. "whatever ill just khanacademy it later". it's become a verb for me and my classmates

Thanks so much Khan! This video made more sense to me in three minutes than my teacher did in an hour!!

In pre calc from now on imma be like whatever's I don't need to pay attention, I'll just 'khan' it later 😉

Great video

Doesn't the exponent (even or odd) relates to determining if the function is even or odd?

hey Youtube!!!! Have my own Math Tutorial Channel, I posted a tutorial on Even and Odd Functions so check it out, Search Thinktanktutor

You are just simply good.

I wish Professors would be forced to watch Khanacademy just to really learn the art of knowledge transfer.

arows must be only to the possitive axes : P

Can't wait for the day where my life will depend on whether or not I can determine whether a function is odd or even. Thrilling stuff.

how we can know if the 2X/X^2-4 is odd or even function

Thanks! best math class ever!

Really helped out with my quiz and homework thank you

Thanxxx … ur so amazing

Im really fascinated of your mouse skills.

And thanks!

I remember even and odd because even functions are even degrees, odd function are odd degrees

Which app are you using? Really a good math video.

You draw better with a mouse then I can with my hand haha

you explain so beautifully you make me want to cry Khan. You a true homie

for it to be even it has to be symmetric only to the y axis, not the x or origin or x=2????

Learned this in class, but found it hard to comprehend the notes my professor gave us, and textbook is rather all over the place and breezes over this. You made this so easy for me to understand. Thank you so much!

For odd functions, does it always have to start from the first quadrant?

I think it is called "Even Function" because every function that it's power is even

, like x^2=f(x), or f(x)= x^4+x^2

Thank you!

Thank you! This helped me a lot!

best teacher ever..thank you so much

You are a genius! Im taking a online algebra course, and they gave me a youtube link to an example of odd and even functions. They started off by showing us the fomula f(x)=f(-x) it is even. I had no clue what he was talking about. By you showing that first example i now completely understand! Way to go Khan!

Thanks, I was looking at my math book and my brain turned to mush until I saw this.

thank you very much

4:25– angry chinese guy

5:30– mustache

thanks khan

Thank you very much.

Thank you! I have a midterm tomorrow… hahaha….

Just to make sure I'm understanding this, y= F(x), and when doing even-odd properties we are always reflecting over the y-axis?

great video

Can you also say in the graph (s) of odd functions the shape from 1st quadrant would have a rotation of 180 degrees about the origin? Because the shape of f (x) = x^3 in 3rd quadrant is same as if you rotate 1st quadrant's shape through 180 degrees about the origin

wouldn't it make more sense to call the even functions "reflected or reflective functions" and the odd functions would be "symmetric functions"?

Um, am I being over simplistic, or do all even functions only have even exponents(x^2, x^4+x^2), and odd functions have only odd exponents(x, x^3, x^5-x^3), while combinations of odd and even or the addition of numbers makes them neither odd nor even?(x^4-x^3, x^2+4)

great video …. thank you dude

thank you!

In 2:59, you said that there is no correlation between even numbers and even functions, but actually even functions are always x^2,X^4,x^6… the power is an even number

I liked it

my actual math teacher hasnt even shown up to class and expected us to learn this. Now i understand after watching this video. Thanks G

Hi. In the video, you say you can't see any correlation between Even + Odd functions and Even/odd numbers. But, while watching this it became evident to me that Odd powers contribute to odd functions and Even powers to Even functions. Could this be the case?

in the second example of odd function he drew, there are two values for f(0) ..so it's not even a function,right?

4:28 am i the only one who sees it as hades????? #illuminati_confirmed

12:23 it sounded like he was gonna say more things, but the video was cut

Hello

Thanks a lot

Can I know what is the program you used in the video to explain even and odd functions?

very good

vid

yayay

Dude you're cool

How do you manage to teach every subject perfectly.

a function f(t) =2piet-t^2 is even or odd??

Sal, you're a god

Perhaps the reason why they are given the name "even" and "odd" functions has to do with the "evenness" or "oddness" of the powers of x in the function? For example, x to any even power will invert the original sign of x, whereas x to any odd power will keep the original sign of x. Therefore, any function with all even powers should be considered even because we know from this rule that it will satisfy f(x) = f(-x) (since all signs are inverted). Likewise, if all powers of x are not even, you know you will have at least one different number because one of the x's will have a different sign, therefore not satisfying f(x) = f(-x) but rather satisfying f(x) = -f(-x) (so called "odd" functions, when a power of x is odd). However, I feel like I'm missing something for functions are are considered neither even or odd…

I'm just a student trying to make sense of this, if anyone wants to jump in and correct this or verify this, I'd appreciate it.

Thank you!

If x(t) is even then why x(t -1) is not even? Please do explain. I will be highly thankful.

Amazing

sum of cos cube x and sin square x will always be odd or even

plz reply

Thank you.

thank you very much,always appreciated your unparalleled teaching

(x^2+sin^2y)^1/2

root(x square +sin square y) is even or odd function

Nice expalanation

this is so helpful, especially when your teacher can’t teach and you have a test tomorrow lmao

Very helpinggggggggggggggggggggggggggggggggggg

Thank you so much

Mr Woods class wya?

Love u bro

The best examples of even and odd functions are COS(x) and SIN(x) respectively. You did get a COS-like wave in there, but never called it 'COS'. Thanks for the best videos!

This helped so much 😭

Amazing buddy

God! You explained way more clearly than my teacher.

You're saying the "even" doesn't necessarily means "even degree", but "The Organic Chemistry Tutor" says that even means even degree. Which of you is right??

Im a 4th grader studying math quiz bee final round

Me:im gonna have a bad time