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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.

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  • leaf grey style avatar for user Aaron
    So what actually happened at ?
    (3 votes)
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  • leaf green style avatar for user Aadya
    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:)
    (5 votes)
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  • aqualine seed style avatar for user Yen Diep
    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?
    (2 votes)
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    • leaf orange style avatar for user A/V
      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.
      (5 votes)
  • ohnoes default style avatar for user Brian Lee
    In the beginning, is the answer is h(x) by odd?
    (3 votes)
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  • mr pants orange style avatar for user ericw74
    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
    (2 votes)
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    • piceratops tree style avatar for user VVCephei
      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.
      (2 votes)
  • aqualine seed style avatar for user Mickey
    Maybe the Autobots are fighting Megatron again. That explains the sound at . Sal, you should probably run.
    (2 votes)
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  • leaf green style avatar for user Pranav
    What exactly does it mean for a function to be odd?
    (1 vote)
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  • leaf blue style avatar for user TB
    Do all odd functions cross the origin?
    (2 votes)
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  • duskpin tree style avatar for user 966worthmoree
    How do I know what the odd of the function is?
    What does the -a(-j) mean?
    (1 vote)
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    • leaf green style avatar for user kubleeka
      A function f(x) is odd if f(-x)=-f(x) for every x. A function is even if f(-x)=f(x) for every x. Functions may be odd, even, both, or neither.

      Sal wrote j(a) as the function j whose input is a. Because Sal said that j was an odd function, that means j(-a)=-j(a), or equivalently, that -j(-a)=j(a).
      (2 votes)
  • starky tree style avatar for user Corin
    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.
    (1 vote)
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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.