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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

• 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.