Understanding Even and Odd Functions: Exponents and Symmetry Tests Explained

[math nerd]

Can you always tell if a function is odd or even by looking at the exponents of each of the variables? My book says you can but when I look in other books it gives examples when that is not true. Or do you always have to do symmetry tests to decide?

 

[HallsofIvy]

It doesn't help to post twice! I will delete the other thread.

The answer to your question is "yes and no"!

IF the function in question is a polynomial then yes: a function is "even" if and only if all powers of x are even. A function is "odd" if and only if all powers of x are odd. That should be easy to remember! I suspect your book is only talking about polynomials.

However, the concepts of "even" and "odd" functions apply to all functions, not just polynomials. A function is called "even" if f(-x)= f(x) for all numbers x (changing the sign on x doesn't change the value of the function at all), odd if f(-x)= -f(x) (changing the sign on x only changes the sign on f(x)) for all numbers x. Of course, with most functions there is no simple relation between f(x) and f(-x): most functions are neither even nor odd.

f(x)=cos(x) for example is an even function while g(x)= sin(x) is an odd function.
f(x)= x+ 3 is neither even nor odd.