# How do even and odd functions relate to even and odd numbers?

How do the notions of oddness and evenness apply to both functions and numbers?

If Even and Odd functions share nothing in common with Even and Odd numbers, then why were Even and Odd adopted for functions? Why not use other adjectives? James Stewart, Calculus 7th ed. 2011. This isn't the Early Transcendentals version.

• $x^n$ is even when $n$ is even and odd when $n$ is odd – Randall Jan 11 at 3:10

For an even function $$f(-x)=f(x).$$ Now you know that $$(-x)^n=x^n$$ if $$n$$ is even.
For an odd function $$g(-x)=-g(x)$$. Now you also know that $$(-x)^n=-x^n$$ if $$n$$ is odd.
2. The Taylor series (at $$x = 0$$) of an even function has only even terms $$x^{2n}$$, and the Taylor series of an odd function has only odd terms $$x^{2n+1}$$. If you don't know what a Taylor series is, we can restrict our attention to polynomials: a polynomial is even iff it has only even terms, and a polynomial is odd iff it has only odd terms.