Even and odd functions: Find the mistake

- [Instructor] We are told Jayden was asked to determine whether f of x is equal to x minus the cube root of x is even, odd, or neither. Here is his work. Is Jayden's work correct? If not, what is the first step where Jayden made a mistake? So pause this video and review Jayden's work, and see if it's correct, or if it's not correct tell me where it's not correct. All right, now let's work this together. So, let's see, just to remind ourselves what Jayden's trying to do, he's trying to decide, whether f of x is even, odd, or neither. And f of x is expressed, or is defined, as x minus the cube root of x. So let's see, the first thing that Jayden did is he's trying to figure out what is f of negative x? Because remember, if f of negative x is equal to f of x, we are even, and if f of negative x is equal to negative f of x, then we are odd. So it makes sense for him to find the expression for f of negative x. So he tries to evaluate f of negative x, and when he does that, everywhere where he sees an x in f of x, he replaces it with a negative x. So that seems good. And then, let's see, this becomes a negative x, that makes sense, minus, and then, a negative x under the radical, and this is a cube root right over here, that's the same thing as negative one times x. The cube root of negative one is negative one. So he takes that negative out of the radical, out of the cube root. So this makes sense, and so then he has a negative x and you subtract a negative, you get a positive. So then that makes sense. And then, the next thing he says is, or he's trying to do, is check if f of negative x is equal to f of x or f of negative x. So he's gonna check whether this is equal to one of them. And so here Jayden says, negative x plus the cube root of x, so that's what f of negative x, what he evaluated it to be, isn't the same as f of x, now let's see is that the case? Is it not the same as f of x? Yup it's definitely, it's not the same as f of x, or negative f of x which is equal to negative x minus the cube root of x. Now that seems a little bit fishy. Did he do the right thing, right over here? Is negative f of x equal to negative x minus the cube root of x? Let's see, negative of f of x is going to be a negative times this entire expression, it's going to be a negative up front, times x minus the cube root of x, and so this is going to be equal to, you distribute the negative sign, you get negative x plus the cube root of x. So Jayden calculated the wrong negative f of x right over here. So, he isn't right that negative x plus the cube root of x, it is actually the same as negative f of x. So he's wrong right over here. So Jayden's mistake is right over here, really it looks like he didn't evaluate negative f of x correctly. So Jayden's work, is Jayden's work correct? No. If not, what is the first step where Jayden made a mistake? Well it would be step two. What he should have said is, it actually is the same as negative f of x, and so therefore his conclusion should be that f of x is odd.