Even and Odd Functions A function f with the property that f(–x) = f(x) for all x is called an even function. Example 1: f(x) = |x| is an even function because f(–x) = |–x| = |x| = f(x). A function f with the property that f(–x) = –f(x) for all x is called an odd function. Example 2: g(x) = 1/x is odd because g(–x) = 1/(–x) = –1/x = –g(x). A polynomial function is a function for which each term has the form axn where a is a real number and n is a positive integer. Polynomial functions with only even powers of x are even functions. Keep in mind that a constant c is the same as cx0 , and so c is an even power of x. Example 3: Here are some examples of polynomial functions that are even. f(x) = x2 g(x) = 4 h(x) = 3x8 – 2x6+ 9 Polynomial functions with only odd powers of x are odd functions. Keep in mind that x is the same as x1, and so x is an odd power of x. Example 4: Here are some examples of polynomial functions that are odd. f(x) = x3 g(x) = x h(x) = 3x11 – 2x5+ 9x A quick graphical analysis of even and odd functions The graph of an even function is symmetrical with respect to the y-axis. This means that the y-axis acts like a “mirror,” and the graph “reflects” across this mirror. The graph of an odd function is symmetrical with respect to the origin. This means that if you rotate the graph 180 degrees (or equivalently, turn it upside down) it will look the same as it did right side up. So another way to determine if f(–x) = f(x) is to graph f in your graphing calculator, and see if the y-axis acts like a mirror (see the function g in the figure above). Another way to determine if f(–x) = –f(x) is to graph f in your graphing calculator, and see if it looks the same upside down (see the function h in the figure above). These two techniques will work for all functions (not just polynomials). Check out the Get 800 collection of test prep books to learn how to apply this information to standardized test questions. If you have any questions on this topic, then please post them in the comments below. Speak to you soon! Comments comments