# How to Memorize Even and Odd Functions?

1. Nov 3, 2015

### basty

How do you memorize the even and odd function?

2. Nov 3, 2015

### Mentallic

Even functions can be reflected in the y-axis without being changed and odd functions can be rotated 180o about the origin without being changed. Is this what you're asking to try and remember?

3. Nov 3, 2015

### tommyxu3

I memorize by that $f(x)=x^n,n\in \mathbb{N}$ is an odd function if $n$ is odd but is an even function if $n$ is even.

4. Nov 3, 2015

### HallsofIvy

Staff Emeritus
I'm not sure what you mean you are "memorizing". It is true that a polynomial that includes only odd powers is odd and a polynomial that includes only even powers is even but there exist many even or odd functions that are not polynomials, such as cos(x) and sin(x). If you mean you are thinking of $x^n$ to remember the definition of "even" and "odd" functions, surely it is not that difficult to remember the definitions themselves.

5. Nov 3, 2015

### tommyxu3

Yes, I know their definitions exactly, and I do just mean the memorization, for I think this may be what the starter meant.
Besides, $f(x)=0$ is also a polynomial and is both odd and even, which didn't only include odd powers, though its degree makes lots of explanations.

6. Nov 3, 2015

### Staff: Mentor

Perhaps sketch a straight line at 45° in the first quadrant. Picture that as one half of the letter V, and complete the other half of the "V" by drawing its mirror image in the y axis.

The letter "V" represents an EVEN FUNCTION.

7. Nov 4, 2015

### rs1n

If $f(-x) = f(x)$ then the function is even. If $f(-x) = -f(x)$ then the function is odd.

Example: $f(x)=2x^4+4x^2-1$ is even since $f(-x) = 2(-x)^4+4(-x)^2-1 = 2x^4+4x^2-1 = f(x)$.

Example: $f(x)=x^3-3x$ is odd since $f(-x) = (-x)^3 - 3(-x) = -x^3 + 3x = -(x^3-3x) = -f(x)$

Exaple: $f(x)=x^2-x$ is neither even nor odd. Note that $f(-x)=(-x)^2-(-x) = x^2+x$ so that $f(-x)\not= f(x)$ (not even) and $f(-x)\not=-f(x)$ (not odd).