Recognizing odd and even functions | Functions and their graphs | Algebra II | Khan Academy

Recognizing odd and even functions | Functions and their graphs | Algebra II | Khan Academy


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.

100 thoughts on “Recognizing odd and even functions | Functions and their graphs | Algebra II | Khan Academy

  • February 27, 2012 at 1:19 am
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    @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.

    Reply
  • March 13, 2012 at 11:42 am
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    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?

    Reply
  • April 3, 2012 at 11:47 pm
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    great video, but i think the most common Odd function would actually be f(x) = x

    Reply
  • April 25, 2012 at 4:38 am
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    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.

    Reply
  • June 6, 2012 at 7:28 pm
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    he has a good voice.

    Reply
  • September 7, 2012 at 3:09 am
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    I have to teach this concept tomorrow; I'm so nervous…

    Reply
  • September 9, 2012 at 11:26 am
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    great job broo….used ur vid for my signals & systems class
    keep it up dude

    Reply
  • September 18, 2012 at 1:43 am
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    Great lecture, I wish my Professor was as proficient in his lectures. Thanks!

    Reply
  • September 18, 2012 at 9:47 am
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    I think I missed it then. Sorry.

    Reply
  • September 25, 2012 at 3:06 am
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    If only every teacher was as clear and understanding as KhanAcademy.

    Reply
  • September 27, 2012 at 1:00 am
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    4:30 looks like a face lol

    Reply
  • October 3, 2012 at 9:16 am
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    Brilliant as usual, thank you.

    Reply
  • October 23, 2012 at 1:07 am
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    hour long lesson in 12 minutes. brilliant.

    Reply
  • November 26, 2012 at 12:47 am
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    Thank you!

    Reply
  • January 5, 2013 at 2:07 pm
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    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.

    Reply
  • January 23, 2013 at 12:14 pm
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    Thank you Guruji

    Reply
  • January 25, 2013 at 3:50 am
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    My teacher should just let us watch khanacdemy on our smartphones it would save everyone timee!

    Reply
  • March 9, 2013 at 2:37 am
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    …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

    Reply
  • March 25, 2013 at 12:25 am
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    no, if its odd its symmetric across the origin not "both axis"

    Reply
  • May 14, 2013 at 11:14 pm
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    thanks man u do to me a great thing

    Reply
  • May 22, 2013 at 3:06 pm
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    Thanks Khan, you're the man!

    Reply
  • August 16, 2013 at 6:12 am
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    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

    Reply
  • August 19, 2013 at 6:23 pm
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    The second even function you drew looks like an angry face XD

    Reply
  • September 12, 2013 at 1:39 pm
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    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!

    Reply
  • September 12, 2013 at 8:23 pm
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    Same here. I feel ya

    Reply
  • September 17, 2013 at 4:02 pm
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    You sir are the reason I pass classes

    Reply
  • September 26, 2013 at 10:59 pm
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    you're so awesome =D

    Reply
  • September 30, 2013 at 5:47 pm
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    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

    Reply
  • October 1, 2013 at 10:31 pm
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    Thanks so much Khan! This video made more sense to me in three minutes than my teacher did in an hour!!

    Reply
  • October 1, 2013 at 11:44 pm
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    In pre calc from now on imma be like whatever's I don't need to pay attention, I'll just 'khan' it later 😉

    Reply
  • October 4, 2013 at 12:24 am
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    Great video

    Reply
  • October 8, 2013 at 7:47 pm
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    Doesn't the exponent (even or odd) relates to determining if the function is even or odd?

    Reply
  • October 18, 2013 at 3:30 am
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    hey Youtube!!!! Have my own Math Tutorial Channel, I posted a tutorial on Even and Odd Functions so check it out, Search Thinktanktutor

    Reply
  • October 21, 2013 at 11:33 pm
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    You are just simply good.
    I wish Professors would be forced to watch Khanacademy just to really learn the art of knowledge transfer.

    Reply
  • October 25, 2013 at 8:19 pm
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    arows must be only to the possitive axes : P

    Reply
  • October 30, 2013 at 2:42 am
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    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.

    Reply
  • December 27, 2013 at 6:18 am
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    how we can know if the 2X/X^2-4 is odd or even function 

    Reply
  • January 14, 2014 at 4:57 am
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    Thanks! best math class ever!

    Reply
  • February 10, 2014 at 5:12 pm
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    Really helped out with my quiz and homework thank you

    Reply
  • February 25, 2014 at 5:16 pm
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    Thanxxx … ur so amazing

    Reply
  • May 8, 2014 at 2:25 am
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    Im really fascinated  of your mouse skills.
    And thanks!

    Reply
  • June 26, 2014 at 11:33 pm
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    I remember even and odd because even functions are even degrees, odd function are odd degrees

    Reply
  • August 15, 2014 at 3:31 am
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    Which app are you using? Really a good math video.

    Reply
  • September 9, 2014 at 5:46 am
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    You draw better with a mouse then I can with my hand haha

    Reply
  • September 17, 2014 at 8:02 am
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    you explain so beautifully you make me want to cry Khan. You a true homie 

    Reply
  • September 30, 2014 at 2:35 am
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    for it to be even it has to be symmetric only to the y axis, not the x or origin or x=2????

    Reply
  • October 23, 2014 at 3:50 am
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    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!

    Reply
  • January 21, 2015 at 9:08 am
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    For odd functions, does it always have to start from the first quadrant?

    Reply
  • February 26, 2015 at 2:38 pm
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    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

    Reply
  • March 2, 2015 at 11:51 am
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    Thank you!

    Reply
  • March 12, 2015 at 12:55 am
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    Thank you! This helped me a lot!

    Reply
  • March 14, 2015 at 8:09 pm
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    best teacher ever..thank you so much

    Reply
  • June 12, 2015 at 8:46 pm
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    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!

    Reply
  • August 22, 2015 at 4:41 pm
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    Thanks, I was looking at my math book and my brain turned to mush until I saw this.

    Reply
  • September 25, 2015 at 8:30 pm
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    thank you very much

    Reply
  • September 27, 2015 at 5:28 am
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    4:25– angry chinese guy

    Reply
  • September 27, 2015 at 5:29 am
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    5:30– mustache

    Reply
  • December 9, 2015 at 1:11 am
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    thanks khan

    Reply
  • December 20, 2015 at 8:55 am
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    Thank you very much.

    Reply
  • February 1, 2016 at 10:59 am
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    Thank you! I have a midterm tomorrow… hahaha….

    Reply
  • April 15, 2016 at 10:08 pm
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    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?

    Reply
  • June 7, 2016 at 12:41 am
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    great video

    Reply
  • July 11, 2016 at 11:06 pm
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    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

    Reply
  • August 16, 2016 at 8:34 pm
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    wouldn't it make more sense to call the even functions "reflected or reflective functions" and the odd functions would be "symmetric functions"?

    Reply
  • September 17, 2016 at 4:45 am
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    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)

    Reply
  • October 21, 2016 at 7:49 pm
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    great video …. thank you dude

    Reply
  • October 30, 2016 at 2:44 pm
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    thank you!

    Reply
  • November 2, 2016 at 8:31 pm
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    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

    Reply
  • November 5, 2016 at 5:50 am
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    I liked it

    Reply
  • February 10, 2017 at 2:21 am
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    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

    Reply
  • March 24, 2017 at 11:30 pm
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    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?

    Reply
  • June 21, 2017 at 12:26 pm
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    in the second example of odd function he drew, there are two values for f(0) ..so it's not even a function,right?

    Reply
  • September 7, 2017 at 2:23 pm
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    4:28 am i the only one who sees it as hades????? #illuminati_confirmed

    Reply
  • September 13, 2017 at 6:26 pm
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    12:23 it sounded like he was gonna say more things, but the video was cut

    Reply
  • September 13, 2017 at 6:36 pm
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    Hello
    Thanks a lot
    Can I know what is the program you used in the video to explain even and odd functions?

    Reply
  • October 22, 2017 at 9:10 am
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    very good
    vid
    yayay

    Reply
  • December 20, 2017 at 3:02 pm
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    Dude you're cool
    How do you manage to teach every subject perfectly.

    Reply
  • January 2, 2018 at 7:02 pm
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    a function f(t) =2piet-t^2 is even or odd??

    Reply
  • January 23, 2018 at 9:45 pm
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    Sal, you're a god

    Reply
  • February 2, 2018 at 1:59 am
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    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.

    Reply
  • February 3, 2018 at 5:42 am
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    Thank you!

    Reply
  • March 20, 2018 at 1:38 pm
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    If x(t) is even then why x(t -1) is not even? Please do explain. I will be highly thankful.

    Reply
  • May 9, 2018 at 5:24 am
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    Amazing

    Reply
  • May 15, 2018 at 6:17 am
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    sum of cos cube x and sin square x will always be odd or even
    plz reply

    Reply
  • June 5, 2018 at 5:04 pm
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    thank you very much,always appreciated your unparalleled teaching

    Reply
  • September 15, 2018 at 7:25 am
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    (x^2+sin^2y)^1/2
    root(x square +sin square y) is even or odd function

    Reply
  • September 23, 2018 at 11:50 am
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    Nice expalanation

    Reply
  • October 15, 2018 at 12:44 am
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    this is so helpful, especially when your teacher can’t teach and you have a test tomorrow lmao

    Reply
  • October 30, 2018 at 1:47 pm
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    Very helpinggggggggggggggggggggggggggggggggggg

    Reply
  • October 31, 2018 at 6:04 pm
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    Thank you so much

    Reply
  • January 10, 2019 at 10:41 pm
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    Mr Woods class wya?

    Reply
  • January 17, 2019 at 4:33 pm
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    Love u bro

    Reply
  • February 3, 2019 at 7:14 am
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    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!

    Reply
  • June 22, 2019 at 7:25 am
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    This helped so much 😭

    Reply
  • August 12, 2019 at 1:05 pm
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    Amazing buddy

    Reply
  • August 29, 2019 at 6:13 am
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    God! You explained way more clearly than my teacher.

    Reply
  • September 14, 2019 at 1:22 am
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    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??

    Reply
  • September 25, 2019 at 10:52 am
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    Im a 4th grader studying math quiz bee final round

    Me:im gonna have a bad time

    Reply

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