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