A number (more specifically, an integer) is even if it is congruent to zero, modulo 2. In other words, when you divide the number by two, you will get a whole number (not a-number-and-a-half), or a number x is even if (and only if) it can be written as x = 2n for some integer n.
If x and y are even numbers, and z is an odd number, then:
- x + y is even
- x * y is even
- x + z is odd
- x * z is even
A function f is even if for every value x in its domain, -x is also in the domain and f(-x) = f(x). A graph of an even function is symmetrical about the y-axis.
If f and g are even functions, and h is an odd function, then:
- nf(x) is even for any real number n
- f(x) + g(x) is even
- f(x) * g(x) is even
- f(x) * h(x) is odd
- f(g(x)) is even
- f(h(x)) is even
- h(f(x)) is even
- If the domains of f and h overlap, then f(x) + h(x) is not even, but it is not odd either unless f(x) = 0
A polynomial of x is even if the power of x in every term is an even number. (Note that (-x)2n = ((-x)2)n = ((x)2)n = (x)2n; every term in an even polynomial is an even function. Also note that this includes the independent term; 0 is even and x0 = 1, which is independent of x.)
Even polynomials are even functions, and continuous even functions can be approximated to any desired accuracy by even polynomials. However, there are functions which are neither even nor odd, and there are polynomials which are neither even nor odd. This is different than for integers; every integer is either even or odd. (In fact, the function f(x) = 0 is both even and odd, thanks to ariels for setting this straight for me.)
BrianShader reminds me: You might like to point out that any function can be written as an even function plus an odd function.