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### Course: 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 have the function f(x)=x^4+x^3, is it even of odd?
• If you use a graphing calculator, it is neither
• Is there any other method than looking at the exponent to find out if it is even or odd function?
• 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).
• Is zero a odd or even number?
• Zero is even and not odd, but f(x) = 0 is both an even and 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.
• 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.
• y = 2 is what kind of function?
• 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
• Would f(x) = 0 be both even and odd?
• When is a function neither even nor odd?!
• 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.
• Can a function be even if it is symmetrical but doesn't cross at 0,0
• 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)
• How can you algebraically determine if a function is even, odd, or neither?