Even and odd functions: Equations | Transformations of functions | Algebra 2 | Khan Academy

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 5 over 3 minus x to the 4th. 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 5 over 3 minus, and everywhere we see an x, we're going to 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 5 over 3 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 1 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 1 over x, and then I could view negative x as the same thing as negative 1 times x.

So we can factor out, or actually say we could take the negative 1 out of the radical. What is the cube root of negative 1? Well, it's negative 1. So we could say minus, we could say minus 1 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 1 over x plus the cube root of x, which is equal to the negative of g of x.

So 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 2 to the negative x plus 2 to the negative of negative x, which would be 2 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... all right, all right, y. Let's evaluate j of negative x is equal to negative x over 1 minus negative x, which is equal to negative x over 1 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.