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  • - [Instructor] We're told this table defines function f.

  • All right.

  • For every x, they give us the corresponding f of x.

  • According to the table, is f even, odd, or neither?

  • So pause this video and see

  • if you can figure that out on your own.

  • All right, now let's work on this together.

  • So let's just remind ourselves the definition

  • of even and odd.

  • One definition that we can think of is that f of x,

  • if f of x is equal to f of negative x,

  • then we're dealing with an even function.

  • And if f of x is equal

  • to the negative of f of negative x,

  • or another way of saying that, if f of negative x.

  • If f of negative x, instead of it being equal to f of x,

  • it's equal to negative f of x.

  • These last two are equivalent.

  • Then in these situations,

  • we are dealing with an odd function.

  • And if neither of these are true,

  • then we're dealing with neither.

  • So what about what's going on over here?

  • So let's see.

  • F of negative seven is equal to negative one.

  • What about f of the negative of negative seven?

  • Well, that would be f of seven.

  • And we see f of seven here is also equal to negative one.

  • So at least in that case and that case,

  • if we think of x as seven,

  • f of x is equal to f of negative x.

  • So it works for that.

  • It also works for negative three and three.

  • F of three is equal to f of negative three.

  • They're both equal to two.

  • And you can see and you can kind of visualize in your head

  • that we have symmetry around the y-axis.

  • And so this looks like an even function.

  • So I will circle that in.

  • Let's do another example.

  • So here, once again, the table defines function f.

  • It's a different function f.

  • Is this function even, odd, or neither?

  • So pause this video and try to think about it.

  • All right, so let's just try a few examples.

  • So here we have f of five is equal to two.

  • F of five is equal to two.

  • What is f of negative five?

  • F of negative five.

  • Not only is it not equal to two,

  • it would have to be equal to two

  • if this was an even function.

  • And it would be equal to negative two

  • if this was an odd function, but it's neither.

  • So we very clearly see just looking at that data point

  • that this can neither be even, nor odd.

  • So I would say neither or neither right over here.

  • Let's do one more example.

  • Once again, the table defines function f.

  • According to the table, is it even, odd, or neither?

  • Pause the video again.

  • Try to answer it.

  • All right, so actually let's just start over here.

  • So we have f of four is equal to negative eight.

  • What is f of negative four?

  • And the whole idea here is I wanna say,

  • okay, if f of x is equal to something,

  • what is f of negative x?

  • Well, they luckily give us f of negative four.

  • It is equal to eight.

  • So it looks like it's not equal to f of x.

  • It's equal to the negative of f of x.

  • This is equal to the negative of f of four.

  • So on that data point alone,

  • at least that data point satisfies it being odd.

  • It's equal to the negative of f of x.

  • But now let's try the other points just to make sure.

  • So f of one is equal to five.

  • What is f of negative one?

  • Well, it is equal to negative five.

  • Once again, f of negative x is equal

  • to the negative of f of x.

  • So that checks out.

  • And then f of zero,

  • well, f of zero is of course equal to zero.

  • But of course if you say

  • what is the negative of f of,

  • if you say what f of negative of zero,

  • well, that's still f of zero.

  • And then if you were to take the negative of zero,

  • that's still zero.

  • So you could view this.

  • This is consistent still with being odd.

  • This you could view as the negative of f of negative zero,

  • which of course is still going to be zero.

  • So this one is looking pretty good that it is odd.

- [Instructor] We're told this table defines function f.

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偶函數和奇函數。代數2|代數2|Khan Academy (Even and odd functions: Tables | Transformations of functions | Algebra 2 | Khan Academy)

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    林宜悉 發佈於 2021 年 01 月 14 日
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