Points, Vectors, and Functions
Even and Odd Functions
Unlike the typical college junior that shows up to their morning class wearing pajamas and their retainer, some functions care what they look in the mirror. These functions, called even or odd functions, have some important properties we can take advantage of later.
Consider the y-axis to be a mirror. A function is even if it looks in the mirror and sees itself exactly as is. In other words, it looks the same when reflected across the y-axis.
For any value of x, the values f(x) and f(-x) must be the same.
In symbols, a function is even if f(x) = f(-x).
To check if a function is even we find f(x) and f(-x) and see if they're the same.
Alternatively, a function is odd if it looks in the mirror and sees itself standing upside-down, like the y-axis is a funhouse mirror.
For any value of x, f(-x) is the upside-down version of f(x). That is, f(x) and f(-x) are negatives of each other.
In symbols, a function is odd if f(-x) = -f(x).
If all the terms involve x raised to even exponents, the polynomial is even. Some examples are
f(x) = x2
f(x) = 6x4-7x2
f(x) = x8 + x6 + x22
If all the exponents are odd, the polynomial is odd. Some examples are
f(x) = x3
f(x) = x9 -6x7
f(x) = x101 + x67
If some of the exponents are odd and some of the exponents are even, the polynomial is neither even nor odd.