Even and odd functions: Tables

- [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.