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### Course: Algebra (all content)>Unit 10

Lesson 36: Introduction to symmetry of functions

# 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
• I did enjoy the car crash, thank you.
• 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 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.