I was hoping someone could show or explain why it is that a function of the form $f(x) = ax^d + bx^e + cx^g+\cdots $ going on for some arbitrary length will be an odd function assuming $d, e, g$ and so on are all odd numbers, and likewise why it will be even if $d, e, g$ and so on are all even numbers. Furthermore, why is it if say, $d$ and $e$ are even but $g$ is odd that $f(x)$ will then become neither even nor odd?
Thanks.
Best Answer
An even function is one for which $f(-x) = f(x)$ for all values of $x$ (e.g. evaluating at -6 is the same as evaluating at 6). If $n$ is an even exponent, then $(-x)^n = (-1)^nx^n = x^n$, since an even number of negative signs will cancel out. If all the exponents are even, then this happens for every term in the polynomial, so the result is the same as the original polynomial.
An odd function is one for which $f(-x) = -f(x)$ for all values of $x$ (i.e. the minus sign factors out). If $n$ is an odd exponent, then $(-x)^n = (-1)^nx^n = -x^n$, since an odd number of negative signs leaves just one negative sign remaining. If all the exponents are odd, then we get: $$ f(-x) = ax^d + bx^e + cx^g + \cdots = -ax^d - bx^e - cx^g - \cdots = -(ax^d + bx^e + cx^g + \cdots) = -f(x). $$
If there is a mixture of odd an even exponents, then neither of these nice properties will hold, so the function will be neither even nor odd.