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## Algebra II (2018 edition)

### Unit 4: Lesson 14

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.
• 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.
• In the beginning, is the answer is h(x) by odd?
• 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.
• Maybe the Autobots are fighting Megatron again. That explains the sound at . Sal, you should probably run.
• What exactly does it mean for a function to be odd?
(1 vote)
• A function f(x) is odd when f(-x)=-f(x) for every x. The graph of an odd function has a particular rotational symmetry, so the graph will look the same after you rotate it 180º about the origin.
• Do all odd functions cross the origin?
• Will an odd function always go through the origin?
(1 vote)
• Yes because they must have symmetry around the origina. Tha's part of the definition of an odd function.