How do you prove that a function is even or odd? Let’s start with some examples. Is this function odd, even or neither. Well you may have some intuition about whether it’s odd, even or neither but often your teacher will want you to actually prove the result.
The way to do that is to start by plugging in –x. f(-x) equals –x minus .04 minus (-x)³. -x³ is well the opposite of x³ is (-x)³ and so that’s going to change this to plus .04x³ and I can factor the minus sign out of the whole thing. I get –(x -.04x³). Of course this is just f(x). For f(-x) to equal –f(x), that’s the very definition of am odd function. So we conclude, f is odd.
Let’s do another example. Here we have g(x) equals 25 over x² plus 9. Again just plug in -x. We get 25 over -x² plus 9. The quantity (-x) ² is the same as x². This is exactly the same as 25 over x² plus 9. And this is exactly g(x). For g(-x) to equal g(x) is the very definition of an even function. So we conclude that g is even.
Remember if you’re proving a function even or odd, always start by plugging in –x and see if you can algebraically manipulate the expression into g(x) or its opposite, to prove whether a function is even or odd.