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  • - [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.

- [Instructor] You've likely heard the concept

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A2 初級

関数の対称性入門|関数の変形|代数学2|カーンアカデミー (Function symmetry introduction | Transformations of functions | Algebra 2 | Khan Academy)

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    林宜悉 に公開 2021 年 01 月 14 日
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