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  • - [Instructor] We are asked

  • are the following functions even, odd, or neither?

  • So pause this video and try to work that out on your own

  • before we work through it together.

  • All right, now let's just remind ourselves

  • of a definition for even and odd functions.

  • One way to think about it is what happens

  • when you take f of negative x?

  • If f of negative x is equal to the function again,

  • then we're dealing with an even function.

  • If we evaluate f of negative x,

  • instead of getting the function,

  • we get the negative of the function,

  • then we're dealing with an odd function.

  • And if neither of these are true it is neither.

  • So let's go to this first one right over here,

  • f of x is equal to five over three minus x to the fourth,

  • and the best way I can think about tackling this

  • is let's just evaluate

  • what f of negative x would be equal to.

  • That would be equal to five over three minus

  • and everywhere we see an x,

  • we're gonna replace that with a negative x,

  • to the fourth power.

  • Now what is negative x to the fourth power?

  • Well if you multiply a negative times a negative

  • times a negative, how many times did I do that?

  • If you take a negative to the fourth power,

  • you're going to get a positive,

  • so that's going to be equal to five over three

  • minus x to the fourth, which is once again equal to f of x

  • and so this first one right over here,

  • f of negative x is equal to f of x, it is clearly even.

  • Let's do another example.

  • So this one right over here, g of x,

  • let's just evaluate g of negative x

  • and at any point, you feel inspired

  • and you didn't figure it out the first time,

  • pause the video again and try to work it out on your own.

  • Well g of negative x is equal to one over negative x

  • plus the cube root of negative x

  • and let's see, can we simplify this any?

  • Well we could rewrite this as the negative of one over x

  • and then yeah, I could view negative x

  • as the same thing as negative one times x

  • and so we can factor out,

  • or I should say we could take the negative one

  • out of the radical.

  • What is the cube root of negative one?

  • Well it's negative one,

  • so we could say minus one times the cube root

  • or we could just say the negative of the cube root of x

  • and then we can factor out a negative,

  • so this is going to be equal to negative of one over x

  • plus the cube root of x,

  • which is equal to the negative of g of x,

  • which is equal to the negative of g of x.

  • And so this is odd,

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

  • or in this case it's g of x,

  • g of negative x is equal to the negative of g of x.

  • Let's do the third one.

  • So here we've got h of x

  • and let's just evaluate h of negative x.

  • h of negative x is equal to two to the negative x

  • plus two to the negative of negative x,

  • which would be two to the positive x.

  • Well this is the same thing as our original h of x.

  • This is just equal to h of x.

  • You just swap these two terms

  • and so this is clearly even.

  • And then last but not least, we have j of x,

  • so let's evaluate j of,

  • why did I write y?

  • Let's evaluate j of negative x

  • is equal to negative x over one minus negative x,

  • which is equal to negative x over one plus x,

  • and let's see, there's no clear way

  • of factoring out a negative

  • or doing something interesting

  • where I get either back to j of x,

  • or I get to negative j of x,

  • so this one is neither

  • and we're done.

- [Instructor] We are asked

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A2 初級

偶数と奇数の関数。方程式|関数の変形|代数2|カーンアカデミー (Even and odd functions: Equations | 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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