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.