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## Algebra II (2018 edition)

### Unit 4: Lesson 14

Introduction to symmetry of functions

# Even/odd functions & numbers

The connection between even and odd functions to even and odd numbers. Created by Sal Khan.

## Want to join the conversation?

• If you are given a function like f(x) = 4x^3 + 5x^3 +4x + 4
and you DO NOT have a graphing calculator, is there an easy
way to test if the function is even or odd, without drawing a
graph?
(70 votes)
• Yes, if you have some experience with even and odd functions to draw from.

It helps also to know that any linear combination of even functions results in a new function that is still even, and likewise for odd functions. So what you can do is look at all the individual terms in a function, and treat each one like its own function, and see if they're either all even or all odd. Let's look at your example:

f(x) = 4x^3 + 5x^3 + 4x + 4

Well, first we can simplify by combining those first two terms into 9x^3:

f(x) = 9x^3 + 4x + 4

Let's take the first term as a new function:

g(x) = 9x^3

The variable is a "pure" x; there isn't any constant added on to it to shift it left or right. On quick inspection, you can see that 9x^3 passes through the origin, because at x=0, 9x^3 is also zero. And as Sal just explained, the odd-numbered power on x suggests that this should be an odd function. The coefficient, as it turns out, doesn't matter. You can think of the 9 as simply scaling the function up by nine times in the vertical direction, but that doesn't change the symmetry of it around the origin. So, 9x^3 is an odd function.

What about the next term?

h(x) = 4x

Well, 4x is the same as 4x^1, so we have an odd power. Again, the coefficient merely scales the function up by 4 times in the vertical direction (it makes the line 4 times steeper), but again you can easily see that the symmetry is preserved. So 4x is also an odd function.

What about the 3rd term:

i(x) = 4

Kind of a strange function, just being a constant. But, 4 is the same as 4x^0 power, because anything to the zero power equals 1. And it's pretty easy to see that if i(x) ALWAYS equals 4, because it's a constant, then i(x) = i(-x), and so i(x) is an even function.

So what do we have:

f(x) =
9x^3 (odd)
+4x (odd)
+4 (even)

They're NOT all odd or all even, so f(x) is neither even nor odd.
(210 votes)
• If you have the function f(x)=x^4+x^3, is it even of odd?
(6 votes)
• If you use a graphing calculator, it is neither
(4 votes)
• I've noticed that if a function comes from up and goes down, or down and goes up, it's odd. is that right?
(3 votes)
• It is (usually) necessary but not sufficient. Here's another example of "necessary but not sufficient": a number can only be prime if it is odd (except for the number 2). But being odd is not enough: lots of odd numbers are not prime. So we would say that for a number to be prime, it is necessary but not sufficient for it to be odd.
(4 votes)
• Is there any other method than looking at the exponent to find out if it is even or odd function?
(4 votes)
• Yes, even functions are symmetric about the y axis, or f(-x) = f(x), and odd functions are symmetric about the origin, or -f(-x) = f(x).
(2 votes)
• Is zero a odd or even number?
(4 votes)
• Zero is even and not odd, but f(x) = 0 is both an even and odd function!
(3 votes)
• I'm confused about one thing. A sine function is an odd function. When you are graphing it and the coefficient of x is negative, you are supposed to bring the negative in the front of the function. Example: f(x)=sin(-x) is equal to f(x)=-sin(x). But what if the function is f(x)=3sin(-x)+4, and you brought the negative to the front since it is an off function, f(x)=-3sin(x)+4. How come it does not get distributed to four? I thought the negative is supposed to be distributed to every term with odd functions. I was practicing online and I got it wrong becuase of it. Can anyone answer? I'm really confused.
(3 votes)
• When you add the 4 to the sine, it shifts the whole function up four units, which means that it no longer meets the definition of an odd function. Just like f(x) = x^3 is an odd function, but f(x) = x^3 +4 is not an odd function.
(2 votes)
• Would f(x) = 0 be both even and odd?
(3 votes)
• When is a function neither even nor odd?!
(2 votes)
• A lot of functions are neither even nor odd. For example, if a function is a polynomial with both odd and even exponents, like "f(x) = x^2 + x^1", then the function is neither odd nor even.

And there are many more examples as well. "f(x) = √x" is another example, as is "f(x) = log(x)", and "f(x) = 3^x", and countless others.

In fact, as it turns out, most functions are neither even nor odd.
(2 votes)
• y = 2 is what kind of function?
(2 votes)
• This function is an even function. And in the spirit of this video that connects "even" and "odd" functions with the parity (whether a number is even/odd) of it's exponents, the function y = 2 is indeed even. That is because y = 2 is equivalent to y = 2x^0 and the number zero has even parity.

Therefor when he shows the function y = x^3 + 2, that function is mixing even and odd exponents; ^3 is odd and ^0 is even.

I noticed this on my own when I was going through college algebra, however outside of my sharing this pattern with other students and professors I have yet to come upon the zero parity connection being made either in personal conversation or in print. Hope this helps.

Ben
(2 votes)
• Can a function be even if it is symmetrical but doesn't cross at 0,0
(2 votes)
• Any function always crosses the point (0, 0). So no, a function cannot be even, or anything, if it doesn't cross (0, 0).
(1 vote)

## Video transcript

In the last video on even and odd functions, I talk about how you shouldn't get confused between even functions and even numbers and odd functions and odd numbers. And I said that there wasn't any obvious connection between the word even function and our notion of even numbers, or any connection between odd functions and odd numbers. And, I was wrong. There actually is a relatively obvious connection, and this was pointed out by the YouTube user Nothias. And the connection, I almost explicitly did it in the last example. When I showed an even function, I showed you x squared. When I showed you an odd function, I showed you x to the third power. When I wanted to show you another odd function, I showed you y is equal to x, or f of x is equal to x to the first power. And so you might start to notice what Nothias pointed out, is that these archetypal or these good examples or these simple examples of even and odd functions, when I just have a very simple x raised to some power, whether the power is even or odd, it's going to tell you whether the function is even or odd. And you want to be very careful here. Not all even or odd functions even have exponents in them. They could be trigonometric functions. They might be some other type of wacky functions. You don't have to have exponents. It's just that these exponents are probably where the motivations for calling these even functions and odd functions came from. And let me just be clear. It's not just also any polynomial-- and even in the last video, when we had x to the third plus 1, this was neither even or odd-- but if you just have the pure x raised to some power, then all of the sudden, the motivations for calling them even and odd start to make sense, because if I have f of x is equal to x to the first power-- that's the same thing as y is equal to x-- this is odd. And it gels with the name because we are also raising it to an odd power. If we have f of x is equal to x squared, we saw in the previous video, this is even. And it gels with the idea that we're raising it to an even power. I could keep going. If it was to the x to the third, that is odd. I could keep going. Let me write it this way. In general, if you have f of x is equal to x to the n, then this is odd-- odd function if n is odd, is an odd number. And this is an even function, if n is even. And I want to make it very clear here. The whole point of this video is just to clarify the motivation for calling them even or odd functions. Not all even functions are going to be of this form here, where it's x raised to some even power. And not all odd functions are going to be. And I also don't want you to be confused that if I have something like x to the third and then I have other stuff past that, and you say, oh, x to the third, that's an odd number. But this is not an odd function. Just when it's just a pure stripped down x to the third or x to the first can you really make that statement. But that really is probably where the motivation comes for naming them even or odd functions. And then the other symmetric functions, even if they don't involve an exponent-- maybe this is some type of trigonometric function-- you're calling it even because you're saying it has the same type of symmetry as, say, x squared or x to an even power. So you group them all together as even functions. And then all of these, even though this may or may not have an exponent in it, it has the same type of symmetries as x raised to an odd power. So that's why call them odd functions. Well, thank you Nothias for pointing that out.