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## Algebra (all content)

### Unit 10: Lesson 36

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? •   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.

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.

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.
• If you have the function f(x)=x^4+x^3, is it even of odd? • I've noticed that if a function comes from up and goes down, or down and goes up, it's odd. is that right? • Is there any other method than looking at the exponent to find out if it is even or odd function? • • 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. • • 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
• • • 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. 