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# Even and odd functions: Graphs

Sal picks the function that is odd among three functions given by their graphs. Created by Sal Khan.

## Want to join the conversation?

• So what actually happened at ?
• sal said a car crashed. I thought he fell off his chair.
• OHH just a CAR CRASH happened. "hoped you enjoyed it."
• I certainly did /j
• Maybe the Autobots are fighting Megatron again. That explains the sound at . Sal, you should probably run.
• I was wondering why the odd and even types of a function don't deal with symmetry over the x-axis? As in Odd asks whether the function is symmetrical with respect to the origin f(-x) =
-f(x), and Even is when the function is symmetrical with respect to the y-axis f(-x) = f(x), but why doesn't this deal with symmetry over the x-axis? Thank you:)
• A curve that is symmetric over the x-axis isn't a function, since it fails the vertical line test.
• I did enjoy the car crash, thank you.
• I am not sure if I understand the odd or even function because it's labeled at x=8 and x=-8. Is it because the function itself of -j(-a) would result in the opposite sign or something?
• Yes, that is the right mindset towards to understanding if the function is odd or even.

For it to be odd:
j(a) = -(j(a))
Rather less abstractly, the function would
both reflect off the y axis and the x axis, and it would still look the same
. So yes, if you were given a point (4,-8), reflecting off the x axis and the y axis, it would output: (-4,8)

For it to be even:
j(a) = j(-a)
Less abstractly, the function reflects off the y-axis and would still look the same as the original, non translated function.
• At , it sounded like a bunch of shopping carts crashing, or glass breaking. h(x) is odd. g(x) is even. f(x) is neither. Remember, if you have a linear equation translated up, down, left, or right, then it is going to be a neither.
• Will an odd function always go through the origin?
• Yes because they must have symmetry around the origina. Tha's part of the definition of an odd function.
• why was y=-x+4 not an odd function, i thought that a function is odd when its exponent is an odd number, please explain
• The function is odd if `f(x) = -f(-x)`. The rule of a thumb might be that if a function doesn't intercepts y at the origin, then it can't be odd, and `y = -x + 4` is shifted up and has y-intercept at 4.

Now, evenness or oddness of functions is connected to the exponents, but the exponent has to be odd on every term. And that `4` is actually `4*x^0`, so it's a term with even exponent. And when you have a mixture of even and odd exponents, then the function as a whole ends up being neither even nor odd.
• I didn't get why f(x) is an even function.
I know because f(-2) equals a positive number - six - that the function isn't odd, but what about the symmetry?
Odd functions don't have symmetry over the y-axis, right? So f(x) should be odd.
• f(x) isn't an even function. It's actually neither odd nor even.

I think you misinterpreted Sal's wording here. We are basically testing a point here (2) to see if it meets the criteria of even or odd. Now, we got f(2) = 2 and f(-2) = -6. Now, this clearly doesn't satisfy the properties of an odd function.

By chance, are you getting confused on his wording when he said "So all I have to do is find even one case that violated this constraint to be odd"? If so, he didn't mean "I have to find *one even case*". He said "I have to find *even one case*". Confusing English, but he just says that one case of something not being true is enough to disprove the fact.

## Video transcript

Which of these functions is odd? And so let's remind ourselves what it means for a function to be odd. So I have a function-- well, they've already used f, g, and h, so I'll use j. So function j is odd. If you evaluate j at some value-- so let's say j of a. And if you evaluate that j at the negative of that value, and if these two things are the negative of each other, then my function is odd. If these two things were the same-- if they didn't have this negative here-- then it would be an even function. So let's see which of these meet the criteria of being odd. So let's look at f of x. So we could pick a particular point. So let's say when x is equal to 2. So we get f of 2 is equal to 2. Now, what is f of negative 2? f of negative 2 looks like it is 6. f of negative 2 is equal to 6. So these aren't the negative of each other. In order for this to be odd, f of negative 2 would have had to be equal to the negative of this, would have had to be equal to negative 2. So f of x is definitely not odd. So all I have to do is find even one case that violated this constraint to be odd. And so I can say it's definitely not odd. Now let's look at g of x. So I could use the same-- let's see, when x is equal to 2, we get g of 2 is equal to negative 7. Now let's look at when g is negative 2. So we get g of negative 2 is also equal to negative 7. So here we have a situation-- and it looks like that's the case for any x we pick-- that g of x is going to be equal to g of negative x. So g of x is equal to g of negative x. It's symmetric around the y-- or I should say the vertical axis-- right over here. So g of x is even, not odd. So which of these functions is odd? Definitely not g of x. So our last hope is h of x. Let's see if h of x seems to meet the criteria. I'll do it in this green color. So if we take h of 1-- and we can look at it even visually. So h of 1 gets us right over here. h of negative 1 seems to get us an equal amount, an equal distance, negative. So it seems to fit for 1. For 2-- well, 2 is at the x-axis. But that's definitely h of 2 is 0. h of negative 2 is 0. But those are the negatives of each other. 0 is equal to negative 0. If we go to, say, h of 4, h of 4 is this negative number. And h of negative 4 seems to be a positive number of the same magnitude. So once again, this is the negative of this. So it looks like this is indeed an odd function. And another way to visually spot an odd function is a function-- it's going to go through the origin, and you could essentially flip it over on both axes. So if you flip this, the right half, over the left half, and then flip that over the horizontal axis, you are going to get this right over here. So you see here we're going up and to the right. Here we're going to go down and to the left. And then you curve right over there. You curve up just like that. But the easiest way to test it is just to do what we did, look at a given x. So for example, when x is equal to 8, h of 8 looks like this number right around 8. h of negative 8 looks like it's pretty close to negative 8. So they seem to be the negative of each other. It sounds like a car crash just happened outside. Anyway, hopefully you enjoyed that. Not the car crash, the math problem.